Lexell's theorem

File:Area of a spherical triangle via Lexell circles.png

Given a fixed base $AB,$ an arc of a great circle on a sphere, and two apex points $C$ and $X$ on the same side of great circle $AB,$ Lexell's theorem holds that the surface area of the spherical triangle $\backslash triangle\; ABX$ is equal to that of $\backslash triangle\; ABC$ if and only if $X$ lies on the small-circle arc $B^*\; C\; A^*\backslash !,$ where $A^*$ and $B^*$ are the points antipodal to $A$ and $B,$ respectively.

As one analog of the planar formula $\backslash text\{area\}\; =\; \backslash tfrac12\; \backslash ,\; \backslash text\{base\}\; \backslash cdot\; \backslash text\{height\}$ for the area of a triangle, the spherical excess $\backslash varepsilon$ of spherical triangle $\backslash triangle\; ABC$ can be computed in terms of the base $c$ (the angular length of arc {{nobr|$AB$)}} and "height" $h\_c$ (the angular distance between the parallel small circles {{nobr|$A^*\; B^*\; C$ and $A\; B\; C^*$):}}{{r|eulermistake}}

:$\backslash sin\; \backslash tfrac12\; \backslash varepsilon\; =\; \backslash tan\; \backslash tfrac12\; c\; \backslash ,\; \backslash tan\backslash tfrac12\; h\_c.$

This formula is based on consideration of a sphere of radius $1$, on which arc length is called angle measure and surface area is called spherical excess or solid angle measure. The angle measure of a complete great circle is $2\backslash pi$ radians, and the spherical excess of a hemisphere (half-sphere) is $2\backslash pi$ steradians, where $\backslash pi$ is the circle constant.

In the limit for triangles much smaller than the radius of the sphere, this reduces to the planar formula.

The small circles $A^*\; B^*\; C$ and $A\; B\; C^*$ each intersect the great circle $AB$ at an angle of $\backslash tfrac12\; \backslash varepsilon.$See {{slink||Stereographic projection}} below for a proof of this.

There are several ways to prove Lexell's theorem, each illuminating a different aspect of the relationships involved.

File:Lexell's proof of Lexell's theorem.png

The main idea in Lexell's {{c.|1777}} geometric proof – also adopted by Eugène Catalan (1843), Robert Allardice (1883), Jacques Hadamard (1901), Antoine Gob (1922), and Hiroshi Maehara (1999) – is to split the triangle $\backslash triangle\; A^*\; B^*\; C$ into three isosceles triangles with common apex at the circumcenter $P$ and then chase angles to find the spherical excess $\backslash varepsilon$ of triangle $\backslash triangle\; ABC.$ In the figure, points $A$ and $B$ are on the far side of the sphere so that we can clearly see their antipodal points and all of Lexell's circle $l.${{r|isosceles}}

Let the base angles of the isosceles triangles $\backslash triangle\; B^*\; C\; P$ (shaded red in the figure), $\backslash triangle\; CA^*\; P$ (blue), and $\backslash triangle\; A^*\; B^*\; P$ (purple) be respectively $\backslash alpha,$ $\backslash beta,$ and $\backslash delta.$ (In some cases $P$ is outside {{nobr|$\backslash triangle\; A^*\; B^*\; C$;}} then one of the quantities $\backslash alpha,\; \backslash beta,\; \backslash delta$ will be negative.) We can compute the internal angles of $\backslash triangle\; ABC$ (orange) in terms of these angles: $\backslash angle\; A\; =\; \backslash pi\; -\; \backslash beta\; -\; \backslash delta$ (the supplement of {{nobr|$\backslash angle\; A^*$)}} and likewise $\backslash angle\; B\; =\; \backslash pi\; -\; \backslash alpha\; -\; \backslash delta,$ and finally $\backslash angle\; C\; =\; \backslash alpha\; +\; \backslash beta.$

By Girard's theorem the spherical excess of $\backslash triangle\; ABC$ is

:$\backslash begin\{align\}$

\varepsilon

&= \angle A + \angle B + \angle C - \pi \\[3mu]

&= (\pi - \beta - \delta) + (\pi - \alpha - \delta) + (\alpha + \beta) - \pi \\[3mu]

&= \pi - 2\delta.

\end{align}

If base $AB$ is fixed, for any third vertex $C$ falling on the same arc of Lexell's circle, the point $P$ and therefore the quantity $\backslash delta$ will not change, so the excess $\backslash varepsilon$ of $\backslash triangle\; ABC,$ which depends only on $\backslash delta,$ will likewise be constant. And vice versa: if $\backslash varepsilon$ remains constant when the point $C$ is changed, then so must $\backslash delta$ be, and therefore $P$ must be fixed, so $C$ must remain on Lexell's circle.

File:Steiner's proof of Lexell's theorem.png

Jakob Steiner (1827) wrote a proof in similar style to Lexell's, also using Girard's theorem, but demonstrating the angle invariants in the triangle $\backslash triangle\; A^*\; B^*\; C$ by constructing a cyclic quadrilateral inside the Lexell circle, using the property that pairs of opposite angles in a spherical cyclic quadrilateral have the same sum.{{r|cyclicquad}}{{r|cyclicproof}}

Starting with a triangle $\backslash triangle\; ABC$, let $l$ be the Lexell circle circumscribing $\backslash triangle\; A^*\; B^*\; C,$ and let $D$ be another point on $l$ separated from $C$ by the great circle $B^*\; A^*\backslash !.$ Let $\backslash alpha\_1\; =\; \backslash angle\; C\; A^*\; B^*\backslash !,$ $\backslash beta\_1\; =\; \backslash angle\; A^*\; B^*\; C,$ $\backslash alpha\_2\; =\; \backslash angle\; B^*\; A^*\; D,$ $\backslash beta\_2\; =\; \backslash angle\; D\; A^*\; B^*\backslash !.$

Because the quadrilateral $\backslash square\; A^*\; D\; B^*\; C$ is cyclic, the sum of each pair of its opposite angles is equal, $\backslash angle\; C\; +\; \backslash angle\; D\; =\; \{\}\backslash !$$\backslash alpha\_1\; +\; \backslash alpha\_2\; +\; \backslash beta\_1\; +\; \backslash beta\_2,$ or rearranged $\backslash alpha\_1\; +\; \backslash beta\_1\; -\; \backslash angle\; C\; =\; \{\}\backslash !$$\backslash angle\; D\; -\; \backslash alpha\_2\; -\; \backslash beta\_2.$

By Girard's theorem the spherical excess $\backslash varepsilon$ of $\backslash triangle\; ABC$ is

:$\backslash begin\{align\}$

\varepsilon

&= \angle A + \angle B + \angle C - \pi \\[3mu]

&= (\pi - \alpha_1) + (\pi - \beta_1) + \angle C - \pi \\[3mu]

&= \pi - (\alpha_1 + \beta_1 - \angle C) \\[3mu]

&= \pi - (\angle D - \alpha_2 - \beta_2).

\end{align}

The quantity $\backslash angle\; D\; -\; \backslash alpha\_2\; -\; \backslash beta\_2$ does not depend on the choice of $C,$ so is invariant when $C$ is moved to another point on the same arc of $l.$ Therefore $\backslash varepsilon$ is also invariant.

Conversely, if $C$ is changed but $\backslash varepsilon$ is invariant, then the opposite angles of the quadrilateral $\backslash square\; A^*\; D\; B^*\; C$ will have the same sum, which implies $C$ lies on the small circle $A^*\; D\; B^*\backslash !.$

Euler in 1778 proved Lexell's theorem analogously to Euclid's proof of Elements I.35 and I.37, as did Victor-Amédée Lebesgue independently in 1855, using spherical parallelograms – spherical quadrilaterals with congruent opposite sides, which have parallel small circles passing through opposite pairs of adjacent vertices and are in many ways analogous to Euclidean parallelograms. There is one complication compared to Euclid's proof, however: The four sides of a spherical parallelogram are the great-circle arcs through the vertices rather than the parallel small circles. Euclid's proof does not need to account for the small lens-shaped regions sandwiched between the great and small circles, which vanish in the planar case.{{harvnb|Euler|1797}}, {{harvnb|Papadopoulos|2014}}, {{harvnb|Atzema|2017}}, {{harvnb|Maehara|Martini|2023}} {{pb}}

Euler's proof differs slightly from the proof presented here in that Euler did not consider spherical parallelograms per se, but instead the parallelogram-like regions bounded by great circle arcs on the two sides and by small-circle arcs on top and bottom. The main idea of the proof is the same, but the lens-shaped regions between the two shapes must be treated slightly differently. A proof using spherical parallelograms proper is found in: {{pb}}

{{cite journal |mode=cs2 |last=

Lebesgue |first=Victor-Amédée |author-link=Victor-Amédée Lebesgue |year=

1855 |title=Démonstration du théorème de Lexell |trans-title=Proof of Lexell's theorem |language=fr |journal=Nouvelles annales de mathématiques |volume=14 |pages=24–26 |url=https://archive.org/details/nouvellesannales1418unse/page/24/ |id={{EuDML|96674}} }}

File:Spherical analog of Elements I.35.png

A lemma analogous to Elements I.35: two spherical parallelograms on the same base and between the same parallels have equal area.

Proof: Let $\backslash square\; ABC\_1D\_1$ and $\backslash square\; ABC\_2D\_2$ be spherical parallelograms with the great circle $m$ (the "midpoint circle") passing through the midpoints of sides $BC\_1$ and $AD\_1$ coinciding with the corresponding midpoint circle in $\backslash square\; ABC\_2D\_2.$ Let $F$ be the intersection point between sides $AD\_2$ and $BC\_1.$ Because the midpoint circle $m$ is shared, the two top sides $C\_1D\_1$ and $C\_2D\_2$ lie on the same small circle $l$ parallel to $m$ and antipodal to a small circle $l^*$ passing through $A$ and $B.$

Two arcs of $l$ are congruent, $D\_1D\_2\; \backslash cong\; C\_1C\_2,$ thus the two curvilinear triangles $\backslash triangle\; BC\_1C\_2$ and $\backslash triangle\; AD\_1D\_2,$ each bounded by $l$ on the top side, are congruent. Each parallelogram is formed from one of these curvilinear triangles added to the triangle $\backslash triangle\; ABF$ and to one of the congruent lens-shaped regions between each top side and $l,$ with the curvilinear triangle $\backslash triangle\; D\_2C\_1F$ cut away. Therefore the parallelograms have the same area. (As in Elements, the case where the parallelograms do not intersect on the sides is omitted, but can be proven by a similar argument.)

Proof of Lexell's theorem: Given two spherical triangles $\backslash triangle\; ABC\_1$ and $\backslash triangle\; ABC\_2$ each with its apex on the same small circle $l$ through points $A^*$ and $B^*\backslash !,$ construct new segments $C\_1D\_1$ and $C\_2D\_2$ congruent to $AB$ with vertices $D\_1$ and $D\_2$ on $l.$ The two quadrilaterals $\backslash square\; ABC\_1D\_1$ and $\backslash square\; ABC\_2D\_2$ are spherical parallelograms, each formed by pasting together the respective triangle and a congruent copy. By the lemma, the two parallelograms have the same area, so the original triangles must also have the same area.

Proof of the converse: If two spherical triangles have the same area and the apex of the second is assumed to not lie on the Lexell circle of the first, then the line through one side of the second triangle can be intersected with the Lexell circle to form a new triangle which has a different area from the second triangle but the same area as the first triangle, a contradiction. This argument is the same as that found in Elements I.39.

File:Lexell's theorem via a Saccheri quadrilateral.png

Another proof using the midpoint circle which is more visually apparent in a single picture is due to Carl Friedrich Gauss (1841), who constructs the Saccheri quadrilateral (a quadrilateral with two adjacent right angles and two other equal angles) formed between the side of the triangle and its perpendicular projection onto the midpoint circle {{nobr|$m,$By perpendicular projection of a point $P$ onto a great circle $g$ we mean the foot of the altitude through $P,$ i.e. the intersection between $g$ and the great circle $h$ which is perpendicular to $g$ and passes through $P.$}} which has the same area as the triangle.{{r|saccheri}}

Let $m$ be the great circle through the midpoints $M\_1$ of $AC$ and $M\_2$ of $BC,$ and let $A\text{'},$ $B\text{'},$ and $C\text{'}$ be the perpendicular projections of the triangle vertices onto $m.$ The resulting pair of right triangles $\backslash triangle\; AA\text{'}M\_1$ and $\backslash triangle\; CC\text{'}M\_1$ (shaded red) have equal angles at $M\_1$ (vertical angles) and equal hypotenuses, so they are congruent; so are the triangles $\backslash triangle\; BB\text{'}M\_2$ and $\backslash triangle\; CC\text{'}M\_2$ (blue). Therefore, the area of triangle $\backslash triangle\; ABC$ is equal to the area of Saccheri quadrilateral $\backslash square\; ABB\text{'}A\text{'},$ as each consists of one red triangle, one blue triangle, and the green quadrilateral $\backslash square\; ABM\_2M\_1$ pasted together. (If $C\text{'}$ falls outside the arc $A\text{'}B\text{'},$ then either the red or blue triangles will have negative signed area.) Because the great circle $m,$ and therefore the quadrilateral $\backslash square\; ABB\text{'}A\text{'},$ is the same for any choice of $C$ lying on the Lexell circle $l,$ the area of the corresponding triangle $\backslash triangle\; ABC$ is constant.

The stereographic projection maps the sphere to the plane. A designated great circle is mapped onto the primitive circle in the plane, and its poles are mapped to the origin (center of the primitive circle) and the point at infinity, respectively. Every circle on the sphere is mapped to a circle or straight line in the plane, with straight lines representing circles through the second pole. The stereographic projection is conformal, meaning it preserves angles.

File:Stereographic proof of Lexell's theorem.png

To prove relationships about a general spherical triangle $\backslash triangle\; ABC,$ without loss of generality vertex $A$ can be taken as the point which projects to the origin. The sides of the spherical triangle then project to two straight segments and a circular arc. If the tangent lines to the circular side at the other two vertices intersect at point $E,$ a planar straight-sided quadrilateral $\backslash square\; ABEC$ can be formed whose external angle at $E$ is the spherical excess $\backslash varepsilon\; =\; \backslash angle\; A\; +\; \backslash angle\; B\; +\; \backslash angle\; C\; -\; \backslash pi$ of the spherical triangle. This is sometimes called the Cesàro method of spherical trigonometry, after crystallographer {{ill|Giuseppe Cesàro|de|Giuseppe Raimondo Pio Cesàro|fr|Giuseppe Raimondo de Cesàro}} who popularized it in two 1905 papers.{{r|cesàro}}

Paul Serret (in 1855, a half century before Cesàro), and independently Aleksander Simonič (2019), used Cesàro's method to prove Lexell's theorem. Let $O$ be the center in the plane of the circular arc to which side $BC$ projects. Then $\backslash square\; OBEC$ is a right kite, so the central angle $\backslash angle\; BOC$ is equal to the external angle at $E,$ the triangle's spherical excess $\backslash varepsilon.$ Planar angle $\backslash angle\; BB^*C$ is an inscribed angle subtending the same arc, so by the inscribed angle theorem has measure $\backslash tfrac12\backslash varepsilon.$ This relationship is preserved for any choice of {{nobr|$C$;}} therefore, the spherical excess of the triangle is constant whenever $C$ remains on the Lexell circle $l,$ which projects to a line through $B^*$ in the plane. (If the area of the triangle is greater than a half-hemisphere, a similar argument can be made, but the point $E$ is no longer internal to the angle {{nobr|$\backslash angle\; BOC.$)}}{{r|stereo}}

File:Lexell's theorem via perimeter of the polar triangle.png

Every spherical triangle has a dual, its polar triangle; if triangle $\backslash triangle\; A\text{'}B\text{'}C\text{'}$ (shaded purple) is the polar triangle of $\backslash triangle\; ABC$ (shaded orange) then the vertices $A\text{'}\backslash !,\; B\text{'}\backslash !,\; C\text{'}$ are the poles of the respective sides $BC,\; CA,\; AB,$ and vice versa, the vertices $A,\; B,\; C$ are the poles of the sides $B\text{'}C\text{'}\backslash !,\; C\text{'}A\text{'}\backslash !,\; A\text{'}B\text{'}\backslash !.$ The polar duality exchanges the sides (central angles) and external angles (dihedral angles) between the two triangles.

Because each side of the dual triangle is the supplement of an internal angle of the original triangle, the spherical excess $\backslash varepsilon$ of $\backslash triangle\; ABC$ is a function of the perimeter $p\text{'}$ of the dual triangle {{nobr|$\backslash triangle\; A\text{'}B\text{'}C\text{'}$:}}

:$\backslash begin\{align\}$

\varepsilon

&= \angle A + \angle B + \angle C - \pi \\[3mu]

&= \bigl(\pi - |B'C'|\bigr) + \bigl(\pi - |C'A'|\bigr) + \bigl(\pi - |A'B'|\bigr) - \pi \\[3mu]

&= 2\pi - p',

\end{align}

where the notation $|PQ|$ means the angular length of the great-circle arc $PQ.$

In 1854 Joseph-Émile Barbier – and independently László Fejes Tóth (1953) – used the polar triangle in his proof of Lexell's theorem, which is essentially dual to the proof by isosceles triangles above, noting that under polar duality the Lexell circle $l$ circumscribing $\backslash triangle\; A^*\; B^*\; C$ becomes an excircle $l\text{'}$ of $\backslash triangle\; A\text{'}B\text{'}C\text{'}$ (incircle of a colunar triangle) externally tangent to side $A\text{'}B\text{'}.${{r|polar}}

If vertex $C$ is moved along $l,$ the side $A\text{'}B\text{'}$ changes but always remains tangent to the same circle $l\text{'}.$ Because the arcs from each vertex to either adjacent touch point of an incircle or excircle are congruent, $A\text{'}T\_B\; \backslash cong\; A\text{'}T\_C$ (blue segments) and $B\text{'}T\_A\; \backslash cong\; B\text{'}T\_C$ (red segments), the perimeter $p\text{'}$ is

:$\backslash begin\{align\}$

p' &= |A'B'| + |B'C'| + |C'A'| \\[3mu]

&= \bigl(|A'T_C| + |B'T_C|\bigr) + |C'B'| + |C'A'| \\[3mu]

&= \bigl(|C'A'| + |A'T_B|\bigr) + \bigl(|C'B'| + |B'T_A|\bigr) \\[3mu]

&= |C'T_B| + |C'T_A|,

\end{align}

which remains constant, depending only on the circle $l\text{'}$ but not on the changing side $A\text{'}B\text{'}.$ Conversely, if the point $C$ moves off of $l,$ the associated excircle $l\text{'}$ will change in size, moving the points $T\_A$ and $T\_B$ both toward or both away from $C\text{'}^*$ and changing the perimeter $p\text{'}$ of $\backslash triangle\; A\text{'}B\text{'}C\text{'}\backslash !$ and thus changing $\backslash varepsilon.$

The locus of points $C$ for which $\backslash varepsilon$ is constant is therefore $l.$

Both Lexell ({{c.|1777}}) and Euler (1778) included trigonometric proofs in their papers, and several later mathematicians have presented trigonometric proofs, including Adrien-Marie Legendre (1800), Louis Puissant (1842), Ignace-Louis-Alfred Le Cointe (1858), and Joseph-Alfred Serret (1862). Such proofs start from known triangle relations such as the spherical law of cosines or a formula for spherical excess, and then proceed by algebraic manipulation of trigonometric identities.{{r|trig}}

The sphere is separated into two hemispheres by the great circle $AB,$ and any Lexell circle through $A^*$ and $B^*$ is separated into two arcs, one in each hemisphere. If the point $X$ is on the opposite arc from $C,$ then the areas of $\backslash triangle\; ABC$ and $\backslash triangle\; ABX$ will generally differ. However, if spherical surface area is interpreted to be signed, with sign determined by boundary orientation, then the areas of triangle $\backslash triangle\; ABC$ and $\backslash triangle\; ABX$ have opposite signs and differ by the area of a hemisphere.

Lexell suggested a more general framing. Given two distinct non-antipodal points $A$ and $B,$ there are two great-circle arcs joining them: one shorter than a semicircle and the other longer. Given a triple $A,B,\; C$ of points, typically $\backslash triangle\; ABC$ is interpreted to mean the area enclosed by the three shorter arcs joining each pair. However, if we allow choice of arc for each pair, then 8 distinct generalized spherical triangles can be made, some with self intersections, of which four might be considered to have the same base $AB.$

{{multiple image

| width = 300

| footer = The eight generalized spherical triangles for vertices {{mvar|A, B, C}}, shown in stereographic projection, with orange and purple shading representing areas of opposite signs

| align = none

| image_style=border:none

| image1 = Generalized spherical triangles 1.svg

| image2 = Generalized spherical triangles 2.svg

}}

These eight triangles do not all have the same surface area, but if area is interpreted to be signed, with sign determined by boundary orientation, then those which differ differ by the area of a hemisphere.{{r|extended-triangles}}

In this context, given four distinct, non-antipodal points $A,$ $B,$ $C,$ and $X$ on a sphere, Lexell's theorem holds that the signed surface area of any generalized triangle $\backslash triangle\; ABC$ differs from that of any generalized triangle $\backslash triangle\; ABX$ by a whole number of hemispheres if and only if $A^*\backslash !,$ $B^*\backslash !,$ $C,$ and $X$ are concyclic.

File:Degenerate lunar case of Lexell's theorem.png

As the apex $C$ approaches either of the points antipodal to the base vertices – say $B^*$ – along Lexell's circle $l,$ in the limit the triangle degenerates to a lune tangent to $l$ at $B^*$ and tangent to the antipodal small circle $l^*$ at $B,$ and having the same excess $\backslash varepsilon$ as any of the triangles with apex on the same arc of $l.$ As a degenerate triangle, it has a straight angle at $A$ (i.e. $\backslash angle\; A\; =\; \backslash pi,$ a half turn) and equal angles $B\; =\; B^*\; =\; \backslash tfrac12\backslash varepsilon.${{harvnb|Steiner|1827}}, {{harvnb|Steiner|1841}}, {{harvnb|Atzema|2017}}

As $C$ approaches $B^*$ from the opposite direction (along the other arc of Lexell's circle), in the limit the triangle degenerates to the co-hemispherical lune tangent to the Lexell circle at $B^*$ with the opposite orientation and angles $\backslash angle\; B\; =\; \backslash angle\; B^\backslash star\; =\; \backslash pi\; -\; \backslash tfrac12\backslash varepsilon.$

The area of a spherical triangle is equal to half a hemisphere (excess {{nobr|$\backslash varepsilon\; =\; \backslash pi$)}} if and only if the Lexell circle $A^*B^*C$ is orthogonal to the great circle $AB,$ that is if arc $A^*B^*$ is a diameter of circle $A^*B^*C$ and arc $AB$ is a diameter of $ABC^*\backslash !.$

In this case, letting $D$ be the point diametrically opposed to $C$ on the Lexell circle $A^*B^*C$ then the four triangles $\backslash triangle\; ABC,$ $\backslash triangle\; BAD,$ $\backslash triangle\; CDA,$ and $\backslash triangle\; DCB$ are congruent, and together form a spherical disphenoid $ABCD$ (the central projection of a disphenoid onto a concentric sphere). The eight points $AA^*BB^*CC^*DD^*$ are the vertices of a rectangular cuboid.{{r|disphenoid}}

{{clear}}

File:Spherical parallelogram.png

A spherical parallelogram is a spherical quadrilateral $\backslash square\; ABCD$ whose opposite sides and opposite angles are congruent {{nobr|($AB\; \backslash cong\; CD,$}} $BC\; \backslash cong\; DA,$ $\backslash angle\; A\; =\; \backslash angle\; C,$ {{nobr|$\backslash angle\; B\; =\; \backslash angle\; D$).}} It is in many ways analogous to a planar parallelogram. The two diagonals $AC$ and $BD$ bisect each-other and the figure has 2-fold rotational symmetry about the intersection point (so the diagonals each split the parallelogram into two congruent spherical triangles, $\backslash triangle\; ABC\; \backslash cong\; \backslash triangle\; CDA$ and {{nobr|$\backslash triangle\; ABD\; \backslash cong\; \backslash triangle\; CDB$);}} if the midpoints of either pair of opposite sides are connected by a great circle $m$, the four vertices fall on two parallel small circles equidistant from it. More specifically, any vertex (say {{nobr|$D$)}} of the spherical parallelogram lies at the intersection of the two Lexell circles ($l\_\{cd\}$ and $l\_\{da\}$) passing through one of the adjacent vertices and the points antipodal to the other two vertices.

As with spherical triangles, spherical parallelograms with the same base and the apex vertices lying on the same Lexell circle have the same area; see {{slink|#Spherical parallelograms}} above. Starting from any spherical triangle, a second congruent triangle can be formed via a (spherical) point reflection across the midpoint of any side. When combined, these two triangles form a spherical parallelogram with twice the area of the original triangle.{{harvnb|Lebesgue|1855}}; {{harvnb|Casey|1889}}, [https://archive.org/details/treatiseonspheri00seri/page/n34/ Def. 17], {{p.|18}}; {{harvnb|Todhunter|Leathem|1901}}, [https://archive.org/details/sphericaltrigono00todh/page/239/ Examples XIX, No. 14], {{p.|239}}

The polar dual to Lexell's theorem, sometimes called Sorlin's theorem after A. N. J. Sorlin who first proved it trigonometrically in 1825, holds that for a spherical trilateral $\backslash triangle\; abc$ with sides on fixed great circles $a,\; b$ (thus fixing the angle between them) and a fixed perimeter $p\; =\; |a|\; +\; |b|\; +\; |c|$ (where $|a|$ means the length of the triangle side {{nobr|$a$),}} the envelope of the third side $c$ is a small circle internally tangent to $a,\; b$ and externally tangent to $c,$ the excircle to trilateral $\backslash triangle\; abc.$ Joseph-Émile Barbier later wrote a geometrical proof (1864) which he used to prove Lexell's theorem, by duality; see {{slink|#Perimeter of the polar triangle}} above.{{harvnb|Todhunter|Leathem|1901}}, [https://archive.org/details/sphericaltrigono00todh/page/154/ § 195], {{p.|154}} {{pb}}

{{ cite journal |mode=cs2 |last1=

Sorlin |first1=A. N. J. |last2=

Gergonne |first2=Joseph Diez |author2-link=Joseph Diez Gergonne |year=

1825 |title=Trigonométrie. Recherches de trigonométrie sphérique |trans-title=Trigonometry. Research on spherical trigonometry

|journal=Annales de Mathématiques Pures et Appliquées |volume=15 |pages=273–304 |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433062736982&seq=303 |id={{EuDML|80036}} }}

This result also applies in Euclidean and hyperbolic geometry: Barbier's geometrical argument can be transplanted directly to the Euclidean or hyperbolic plane.

File:Foliation of the sphere by Lexell circles.png

Lexell's loci for any base $AB$ make a foliation of the sphere (decomposition into one-dimensional leaves). These loci are arcs of small circles with endpoints at $A^*$ and $B^*\backslash !,$ on which any intermediate point $C$ is the apex of a triangle $ABC$ of a fixed signed area. That area is twice the signed angle between the Lexell circle and the great circle $ABA^*B^*$ at either of the points $A^*$ or {{nobr|$B^*$;}} see {{slink|#Lunar degeneracy}} above. In the figure, the Lexell circles are in green, except for those whose triangles' area is a multiple of a half hemisphere, which are black, with area labeled; see {{slink|#Half-hemisphere area}} above.{{harvnb|Papadopoulos|Su|2017}}

These Lexell circles through $A^*$ and $B^*$ are the spherical analog of the family of Apollonian circles through two points in the plane.

In 1784 Nicolas Fuss posed and solved the problem of finding the triangle $\backslash triangle\; ABC$ of maximal area on a given base $AB$ with its apex $C$ on a given great circle $g.$ Fuss used an argument involving infinitesimal variation of $C,$ but the solution is also a straightforward corollary of Lexell's theorem: the Lexell circle $A^*B^*C$ through the apex must be tangent to $g$ at $C.$

If $g$ crosses the great circle through $AB$ at a point $P$, then by the spherical analog of the tangent–secant theorem, the angular distance $PC$ to the desired point of tangency satisfies

:$\backslash tan^2\; \backslash tfrac12|PC|\; =\; \backslash tan\; \backslash tfrac12\; |PA^*|\backslash ,\backslash tan\; \backslash tfrac12\; |PB^*|,$

from which we can explicitly construct the point $C$ on $g$ such that $\backslash triangle\; ABC$ has maximum area.{{harvnb|Papadopoulos|2014}}, {{harvnb|Atzema|2017}} {{pb}}

{{cite journal |mode=cs2 |last=

Fuss |first=Nicolas |author-link=Nicolas Fuss |year=1788 |orig-year=written 1784 |title=Problematum quorundam sphaericorum solutio |journal=Nova Acta Academiae Scientiarum Imperialis Petropolitanae |volume=2 |pages=70–83 |url=https://archive.org/details/novaactaacademia02impe/page/70/ }}

In 1786 Theodor von Schubert posed and solved the problem of finding the spherical triangles of maximum and minimum area of a given base and altitude (the spherical length of a perpendicular dropped from the apex to the great circle containing the base); spherical triangles with constant altitude have their apex on a common small circle (the "altitude circle") parallel to the great circle containing the base. Schubert solved this problem by a calculus-based trigonometric approach to show that the triangle of minimal area has its apex at the nearest intersection of the altitude circle and the perpendicular bisector of the base, and the triangle of maximal area has its apex at the far intersection. However, this theorem is also a straightforward corollary of Lexell's theorem: the Lexell circles through the points antipodal to the base vertices representing the smallest and largest triangle areas are those tangent to the altitude circle. In 2019 Vincent Alberge and Elena Frenkel solved the analogous problem in the hyperbolic plane.{{harvnb|Atzema|2017}} {{pb}}

{{cite journal |mode=cs2 |last=

Schubert |first=Friedrich Theodor |author-link=Theodor von Schubert |year=

1789 |orig-date=written 1786 |title=Problematis cuiusdam sphaerici solutio |trans-title=The solution of a certain spherical problem |language=la |journal=Nova Acta Academiae Scientiarum Imperialis Petropolitanae |volume=4 |pages=89–94 |url=https://books.google.com/books?id=FQFlAAAAcAAJ&pg=PA89 }} {{pb}}

{{cite book |mode=cs2 |last1=

Alberge |first1=Vincent |last2=

Frenkel |first2=Elena |year=

2019 |chapter=3. On a problem of Schubert in hyperbolic geometry |title=Eighteen Essays in Non-Euclidean Geometry |editor1-last=Alberge |editor1-first=Vincent |editor2-last=Papadopoulos |editor2-first=Athanase |publisher=European Mathematical Society |pages=27–46 |doi=10.4171/196-1/2 }}

File:Steiner's theorem on area bisectors of a spherical triangle.png

In the Euclidean plane, a median of a triangle is the line segment connecting a vertex to the midpoint of the opposite side. The three medians of a triangle all intersect at its centroid. Each median bisects the triangle's area.

On the sphere, a median of a triangle can also be defined as the great-circle arc connecting a vertex to the midpoint of the opposite side. The three medians all intersect at a point, the central projection onto the sphere of the triangle's extrinsic centroid – that is, centroid of the flat triangle containing the three points if the sphere is embedded in 3-dimensional Euclidean space. However, on the sphere the great-circle arc through one vertex and a point on the opposite side which bisects the triangle's area is, in general, distinct from the corresponding median.

Jakob Steiner used Lexell's theorem to prove that these three area-bisecting arcs (which he called "equalizers") all intersect in a point, one possible alternative analog of the planar centroid in spherical geometry. (A different spherical analog of the centroid is the apex of three triangles of equal area whose bases are the sides of the original triangle, the point with $\backslash bigl(\backslash tfrac13,\backslash tfrac13,\backslash tfrac13\backslash bigr)$ as its spherical area coordinates.){{harvnb|Steiner|1827}}, {{harvnb|Steiner|1841}}, {{harvnb|Atzema|2017}}. {{harvnb|Simonič|2019}} includes another proof of this theorem without relying on Lexell's theorem.

File:Spherical area coordinates.png

The barycentric coordinate system for points relative to a given triangle in affine space does not have a perfect analogy in spherical geometry; there is no single spherical coordinate system sharing all of its properties. One partial analogy is spherical area coordinates for a point $P$ relative to a given spherical triangle $\backslash triangle\; ABC,$

:$\backslash left($

\frac{\varepsilon_{PBC}}{\varepsilon_{ABC}},

\frac{\varepsilon_{APC}}{\varepsilon_{ABC}},

\frac{\varepsilon_{ABP}}{\varepsilon_{ABC}}

\right),

where each quantity $\backslash varepsilon\_\{QRS\}$ is the signed spherical excess of the corresponding spherical triangle $\backslash triangle\; QRS.$ These coordinates sum to $1,$ and using the same definition in the plane results in barycentric coordinates.

By Lexell's theorem, the locus of points with one coordinate constant is the corresponding Lexell circle. It is thus possible to find the point corresponding to a given triple of spherical area coordinates by intersecting two small circles.

Using their respective spherical area coordinates, any spherical triangle can be mapped to any other, or to any planar triangle, using corresponding barycentric coordinates in the plane. This can be used for polyhedral map projections; for the definition of discrete global grids; or for parametrizing triangulations of the sphere or texture mapping any triangular mesh topologically equivalent to a sphere.{{r|sac}}

File:Lexell's theorem in the plane.png

The analog of Lexell's theorem in the Euclidean plane comes from antiquity, and can be found in Book I of Euclid's Elements, propositions 37 and 39, built on proposition 35. In the plane, Lexell's circle degenerates to a straight line (which could be called Lexell's line) parallel to the base.{{r|euclid}}

File:Euclid's Elements I.35.png

Elements I.35 holds that parallelograms with the same base whose top sides are colinear have equal area. Proof: Let the two parallelograms be $\backslash square\; ABC\_1D\_1$ and $\backslash square\; ABC\_2D\_2,$ with common base $AB$ and $C\_1,$ $D\_1,$ $C\_2,$ and $D\_2$ on a common line parallel to the base, and let $F$ be the intersection between $BC\_1$ and $AD\_2.$ Then the two top sides are congruent $C\_1D\_1\; \backslash cong\; C\_2D\_2$ so, adding the intermediate segment to each, $C\_1C\_2\; \backslash cong\; D\_1D\_2.$ Therefore the two triangles $\backslash triangle\; BC\_1C\_2$ and $\backslash triangle\; AD\_1D\_2$ have matching sides so are congruent. Now each of the parallelograms is formed from one of these triangles, added to the triangle $\backslash triangle\; ABF$ with the triangle $\backslash triangle\; D\_2C\_1F$ cut away, so therefore the two parallelograms $\backslash square\; ABC\_1D\_1$ and $\backslash square\; ABC\_2D\_2$ have equal area.

Elements I.37 holds that triangles with the same base and an apex on the same line parallel to the base have equal area. Proof: Let triangles $\backslash triangle\; ABC\_1$ and $\backslash triangle\; ABC\_2$ each have its apex on the same line $l$ parallel to the base $AB.$ Construct new segments $C\_1D\_1$ and $C\_2D\_2$ congruent to $AB$ with vertices $D\_1$ and $D\_2$ on $l.$ The two quadrilaterals $\backslash square\; ABC\_1D\_1$ and $\backslash square\; ABC\_2D\_2$ are parallelograms, each formed by pasting together the respective triangle and a congruent copy. By I.35, the two parallelograms have the same area, so the original triangles must also have the same area.

Elements I.39 is the converse: two triangles of equal area on the same side of the same base have their apexes on a line parallel to the base. Proof: If two triangles have the same base and same area and the apex of the second is assumed to not lie on the line parallel to the base (the "Lexell line") through the first, then the line through one side of the second triangle can be intersected with the Lexell line to form a new triangle which has a different area from the second triangle but the same area as the first triangle, a contradiction.

In the Euclidean plane, the area $\backslash varepsilon$ of triangle $\backslash triangle\; ABC$ can be computed using any side length (the base) and the distance between the line through the base and the parallel line through the apex (the corresponding height). Using point $C$ as the apex, and multiplying both sides of the traditional identity by $\backslash tfrac12$ to make the analogy to the spherical case more obvious, this is:

:$\backslash tfrac12\; \backslash varepsilon\; =\; \backslash tfrac12\; c\backslash ,\backslash tfrac12\; h\_c.$

The Euclidean theorem can be taken as a corollary of Lexell's theorem on the sphere. It is the limiting case as the curvature of the sphere approaches zero, i.e. for spherical triangles as which are infinitesimal in proportion to the radius of the sphere.

File:Lexell's theorem in the hyperbolic plane (half-plane model).png

In the hyperbolic plane, given a triangle $\backslash triangle\; ABC,$ the locus of a variable point $X$ such that the triangle $\backslash triangle\; ABX$ has the same area as $\backslash triangle\; ABC$ is a hypercycle passing through the points antipodal to $A$ and $B,$ which could be called Lexell's hypercycle. Several proofs from the sphere have straightforward analogs in the hyperbolic plane, including a Gauss-style proof via a Saccheri quadrilateral by Barbarin (1902) and Frenkel & Su (2019), an Euler-style proof via hyperbolic parallelograms by Papadopoulos & Su (2017), and a Paul Serret-style proof via stereographic projection by Shvartsman (2007).{{r|hyperbolic}}

In spherical geometry, the antipodal transformation takes each point to its antipodal (diametrically opposite) point. For a sphere embedded in Euclidean space, this is a point reflection through the center of the sphere; for a sphere stereographically projected to the plane, it is an inversion across the primitive circle composed with a point reflection across the origin (or equivalently, an inversion in a circle of imaginary radius of the same magnitude as the radius of the primitive circle).

In planar hyperbolic geometry, there is a similar antipodal transformation, but any two antipodal points lie in opposite branches of a double hyperbolic plane. For a hyperboloid of two sheets embedded in Minkowski space of signature $(-,\; +,\; +),$ known as the hyperboloid model, the antipodal transformation is a point reflection through the center of the hyperboloid which takes each point onto the opposite sheet; in the conformal half-plane model it is a reflection across the boundary line of ideal points taking each point into the opposite half-plane; in the conformal disk model it is an inversion across the boundary circle, taking each point in the disk to a point in its complement. As on the sphere, any generalized circle passing through a pair of antipodal points in hyperbolic geometry is a geodesic.{{r|hyperbolic-antipodes}}

Analogous to the planar and spherical triangle area formulas, the hyperbolic area $\backslash varepsilon$ of the triangle can be computed in terms of the base $c$ (the hyperbolic length of arc {{nobr|$AB$)}} and "height" $h\_c$ (the hyperbolic distance between the parallel hypercycles {{nobr|$A^*\; B^*\; C$ and $A\; B\; C^*$):}}

:$\backslash sin\; \backslash tfrac12\; \backslash varepsilon\; =\; \backslash tanh\backslash tfrac12\; c\; \backslash ,\; \backslash tanh\backslash tfrac12\; h\_c.$

As in the spherical case, in the small-triangle limit this reduces to the planar formula.

{{reflist|30em|refs=

{{cite journal |mode=cs2 |last=

Cesàro |first=Giuseppe |author-link=Giuseppe Cesàro |year=

1905 |title=Nouvelle méthode pour l'établissement des formules de la trigonométrie sphérique |trans-title=New method for establishing the formulas of spherical trigonometry |language=fr |journal=Académie royale de Belgique: Bulletins de la Classe des sciences |series=ser. 4 |volume=7 |number=9–10 |pages=434–454 |url=https://www.biodiversitylibrary.org/item/310817#page/452/mode/1up }} {{pb}}

{{cite journal |mode=cs2 |last=

Cesàro |first=Giuseppe |year=

1905 |title=Les formules de la trigonométrie sphérique déduites de la projection stéréographique du triangle. – Emploi de cette projection dans les recherches sur la sphère |trans-title=The formulas of spherical trigonometry deduced by spherical projection of the triangle. – Use of this projection in researches on the sphere |language=fr |journal=Académie royale de Belgique: Bulletins de la Classe des sciences |series=ser. 4 |volume=7 |number=12 |pages=560–584 |url=https://www.biodiversitylibrary.org/item/310817#page/582/ }} {{pb}}

{{cite book |mode=cs2 |last=

Donnay |first=Joseph Desire Hubert |author-link=J. D. H. Donnay |year=

1945 |title=Spherical Trigonometry after the Cesàro Method |location=New York |publisher=Interscience |url=https://archive.org/details/bwb_P8-ABB-831/ |url-access=limited }} {{pb}}

{{cite book |mode=cs2 |last=

Van Brummelen |first=Glen |author-link=Glen Van Brummelen |year=

2012 |chapter=8. Stereographic Projection |title=Heavenly Mathematics |publisher=Princeton University Press |pages=129–150 }}

{{harvnb|Papadopoulos|2014}}, {{harvnb|Atzema|2017}}, {{harvnb|Maehara|Martini|2023}} {{pb}}

{{cite journal |mode=cs2 |last=

Steiner |first=Jakob |author-link=Jakob Steiner |year=

1827 |title=Verwandlung und Theilung sphärischer Figuren durch Construction |trans-title=Transformation and Division of Spherical Figures by Construction |language=de |journal=Journal für die reine und angewandte Mathematik |volume=2 |issue=1 |pages=45–63 |url=https://archive.org/details/journalfurdierei1218unse/page/45 |doi=10.1515/crll.1827.2.45 |id={{EuDML|183090}}

}} {{pb}}

{{cite journal |mode=cs2 |last=

Steiner |first=Jakob |display-authors=0 |year=

1845 |title=Théorème de Lexell, et transformation des polygones sphériques, d'après M. Steiner |trans-title=Lexell's theorem, and transformation of spherical polygons, after Mr. Steiner |journal=Nouvelles Annales de Mathématiques |volume=4 |pages=587–590 |language=fr |url=https://hdl.handle.net/2027/pst.000067416042?urlappend=%3Bseq=595 |id={{EuDML|95439}} }} {{pb}}

{{cite journal |mode=cs2 |last=

Steiner |first=Jakob |author-link=Jakob Steiner |year=

1841 |title=Sur le maximum et le minimum des figures dans le plan, sur la sphère et dans l'espace général |trans-title=On the maximum and the minimum of figures in the plane, on the sphere and in general space |language=fr |journal=Journal de mathématiques pures et appliquées |volume=6 |pages=105–170 |url=https://archive.org/details/s1journaldemat06liou/page/105/ |id={{EuDML|234575}} }}

This property was first proven in: {{pb}}

{{cite journal |mode=cs2 |last=

Lexell |first=Anders Johan |author-link=Anders Johan Lexell |year=

1786 |title=De proprietatibus circulorum in superficie sphaerica descriptorum |trans-title=On the properties of circles described on a spherical surface |language=la |journal=Acta Academiae Scientiarum Imperialis Petropolitanae |volume=6: 1782 |issue=1 |pages=58–103, [https://archive.org/details/actaacademiaesci82impe/page/n481/mode/1up figures tab. 3] |url=https://archive.org/details/actaacademiaesci82impe/page/58/ }}

{{cite journal |mode=cs2 |last1=

Maehara |first1=Hiroshi |last2=

Martini |first2=Horst |year=

2017 |title=On Lexell's Theorem |journal=American Mathematical Monthly |volume=124 |number=4 |pages=337–344 |doi=10.4169/amer.math.monthly.124.4.337 }} {{pb}}

{{cite journal |mode=cs2 |last1=

Brooks |first1=Jeff |last2=

Strantzen |first2=John |year=

2005 |title=Spherical Triangles of Area {{mvar|π}} and Isosceles Tetrahedra |journal=Mathematics Magazine |volume=78 |number=4 |pages=311–314 |jstor=30044179 |doi=10.1080/0025570X.2005.11953347 |url=https://www.maa.org/sites/default/files/3004417919846.pdf.bannered.pdf }}

See {{harvp|Papadopoulos|2014}} and {{harvp|Atzema|2017}} about the early history, and {{harvp|Maehara|Martini|2023}} for a variety of proofs. For further context, see: {{pb}}

{{cite book |mode=cs2 |last=

Chasles |first=Michel |author-link=Michel Chasles |year=

1837 |title=Aperçu historique sur l'origine et le développment des méthodes en géométrie |trans-title=Historical overview of the origin and development of methods in geometry |place=Brussels |publisher=Hayez |language=fr |at=Ch. 5, {{nobr|§§ 42–45}}, "Géométrie de la sphère" [Spherical geometry], {{pgs|235–240}} |url=https://archive.org/details/aperuhistoriqu00chasuoft/page/235/}}

Euclid (c. 300 BCE), Elements,

[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI35.html Prop. I.35]: "Parallelograms which are on the same base and in the same parallels equal one another."

[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html Prop. I.37]: "Triangles which are on the same base and in the same parallels equal one another."

[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI39.html Prop. I.39]: "Equal triangles which are on the same base and on the same side are also in the same parallels."

{{harvp|Puissant|1842}} expresses this in terms of the radius of Lexell's circle, as does {{harvp|Euler|1797}} who mistakenly writes $\backslash tan\; \backslash tfrac12\; \backslash varepsilon$ instead of $\backslash sin\; \backslash tfrac12\; \backslash varepsilon.$ {{pb}}

If $r\_c$ is the radius of Lexell's circle, $$\backslash tan\backslash tfrac12\; h\_c\; =\; 1\; /\; \backslash tan\; r\_c.$$ {{pb}}

Note that $h\_c$ is the shortest angular distance from $C$ to the small circle $ABC^*\backslash !,$ not the shortest distance from $C$ to the great circle $AB.$

{{harvnb|Lexell|1784}}, [https://archive.org/details/actaacademiaesci81impe/page/124/ § 11], {{pgs|124–145}}; [https://17centurymaths.com/contents/euler/lexellone.pdf#page=17 in Stén's translation] {{pgs|17–18}} {{pb}}

For more about generalized triangles, see {{harvp|Todhunter|Leathem|1901}}, [https://archive.org/details/sphericaltrigono00todh/page/240/ Ch. 19. "The Extended Definition of the Spherical Triangle"], {{pgs|240–258}} {{pb}}

{{cite book |mode=cs2 |last=

Study |first=Eduard |author-link=Eduard Study |year=

1893 |title=Sphärische trigonometrie, orthogonale substitutionen und elliptische functionen |language=de |trans-title=Spherical trigonometry, orthogonal substitutions and elliptic functions |publisher=S. Hirzel |url=https://hdl.handle.net/2027/coo.31924059019046 }} {{pb}}

{{cite conference |mode=cs2 |last=

Study |first=Eduard |author-link=Eduard Study |year=

1896 |publication-date=|title=Some Researches in Spherical Trigonometry |pages=382–394 |url=https://archive.org/details/cu31924062544352/page/382/ |conference=International Mathematical Congress, Chicago, 1893 |book-title=Mathematical Papers Read at the International Mathematical Congress|publisher=MacMillan }}

Proof by Saccheri quadrilateral: {{pb}}

{{cite book |mode=cs2 |last=

Barbarin |first=Paul Jean Joseph |author-link=Paul Jean Joseph Barbarin |year=

1902 |title=La géométrie non Euclidienne |language=fr |trans-title=Non-Euclidean Geometry |chapter=§ 6.23 Aires planes, triangle et polygone |trans-chapter=Plane areas, triangle and polygon |publisher=Scientia |chapter-url=https://books.google.com/books?id=JYrTAdO34CoC&pg=PA50 |pages=50–55 }} {{pb}}

{{cite book |mode=cs2 |last1=

Frenkel |first1=Elena |last2=

Su |first2=Weixu |year=

2019 |chapter=2. The area formula for hyperbolic triangles |title=Eighteen Essays in Non-Euclidean Geometry |editor1-last=Alberge |editor1-first=Vincent |editor2-last=Papadopoulos |editor2-first=Athanase |publisher=European Mathematical Society |pages=27–46 |doi=10.4171/196-1/2 }} {{pb}}

Euclid-style proof by parallelograms, and a trigonometric proof: {{pb}}

{{cite book |mode=cs2 |last1=

Papadopoulos |first1=Athanase |last2=

Su |first2=Weixu |year=

2017 |contribution=On hyperbolic analogues of some classical theorems in spherical geometry |title= Hyperbolic Geometry and Geometric Group Theory |editor1-last=Fujiwara |editor1-first=Koji |editor2-last=Kojima |editor2-first=Sadayoshi |editor3-last=Ohshika |editor3-first=Ken'ichi |publisher=Mathematical Society of Japan |pages=225–253 |arxiv=1409.4742 |doi=10.2969/aspm/07310225 }} {{pb}}

Proof by stereographic projection with one vertex at the origin: {{pb}}

{{cite journal |mode=cs2 |last=

Shvartsman |first=Osip Vladimirovich |year=

2007 |script-title=ru:Комментарий к статье П. В. Бибикова и И. В. Ткаченко «О трисекции и бисекции треугольника на плоскости Лобачевского» |language=ru |trans-title=Comment on the article by P. V. Bibikov and I. V. Tkachenko 'On trisection and bisection of a triangle in the Lobachevsky plane' |journal=Matematicheskoe Prosveschenie |series=ser. 3 |volume=11 |pages=127–130 |url=https://www.mccme.ru/free-books/matpros/pdf/mp-11.pdf#page=127 }}

{{wikicite |ref = {{harvid|Akopyan|2009}}

|reference = {{cite journal |mode=cs2 |ref=none |last=

Akopyan |first=Arseniy V. |year=

2009 |script-title=ru:О некоторых классических конструкциях в геометрии Лобачевского |language=ru |journal=Matematicheskoe Prosveshenie |series=ser. 3 |volume=13 |pages=155–170 |url=https://www.mccme.ru/free-books/matpros/mpd.pdf#page=155 }}, translated as [https://arxiv.org/abs/1105.2153 "On some classical constructions extended to hyperbolic geometry"], translated by Russell, Robert A., 2011, {{arXiv|1105.2153}}

}} {{pb}}

For a fuller elaboration of antipodal transformations in general, which Norman Johnson calls central inversions, see: {{pb}}

{{cite book |mode=cs2 |last=

Johnson |first=Norman W. |author-link=Norman Johnson (mathematician) |year=

1981 |chapter=Absolute Polarities and Central Inversion |title=The Geometric Vein: The Coxeter Festschrift |editor1-last=Davis |editor1-first=Chandler |editor2-last=Grünbaum |editor2-first=Branko |editor3-last=Sherk |editor3-first=F.A. |publisher=Springer |pages=443–464 |chapter-url=https://archive.org/details/geometricveincox0000unse/page/443/ |chapter-url-access=limited |doi=10.1007/978-1-4612-5648-9_28 }}

{{harvnb|Lexell|1784}}, {{harvnb|Atzema|2017}}, {{harvnb|Maehara|Martini|2023}} {{pb}}

The same basic idea was used in: {{pb}}

{{cite book |mode=cs2 |last=

Catalan |first=Eugène Charles |author-link=Eugène Charles Catalan |year=

1843 |title=Éléments de géométrie |trans-title=Elements of geometry |language=fr |publisher=Bachelier |chapter=Livre 7, Problème 7. Quel est le lieu géométrique des sommets des triangles sphériques de méme base et de méme surface? |trans-chapter=What is the locus of the apices of spherical triangles with the same base and the same area?

|chapter-url=https://babel.hathitrust.org/cgi/pt?id=uva.x001126751&view=1up&seq=321 |pages=271–272 }} {{pb}}

{{cite journal |mode=cs2 |last=

Allardice |first=Robert Edgar |author-link=Robert Edgar Allardice |year=

1883 |title=Spherical Geometry |journal=Proceedings of the Edinburgh Mathematical Society |volume=2 |pages=8–16 |doi=10.1017/S0013091500037020 |doi-access=free }} {{pb}}

{{cite book |mode=cs2 |last=

Hadamard |first=Jacques |author-link=Jacques Hadamard |year=

1901 |title=Leçons de géométrie élémentaire |language=fr |trans-title=Lessons in elementary geometry |volume=2: Géométrie dans l'espace [Geometry in space] |publisher=Armand Colin |chapter=§ 697. Théorème de Lexell. |chapter-url=https://babel.hathitrust.org/cgi/pt?id=msu.31293001875354&seq=424 |pages=392–393 }} {{pb}}

{{cite journal |mode=cs2 |last=

Gob |first=Antoine |author-link=Antoine Gob |year=

1922 |title=Notes de géometrie et de trigonométrie spheriques |language=fr |trans-title=Notes on geometry and spherical trigonometry |journal=Mémoires de la Société Royale des Sciences de Liège |series=ser. 3 |volume=11 |at=No. 3 (pp. 1–29) |url=https://archive.org/details/memoiresdelasoci3111soci/page/n340/ }} {{pb}}

{{cite journal |mode=cs2 |last=

Maehara |first=Hiroshi |year=

1999 |title=Lexell's theorem via an inscribed angle theorem |journal=American Mathematical Monthly |volume=106 |number=4 |pages=352–353 |doi=10.1080/00029890.1999.12005052 }}

{{harvnb|Maehara|Martini|2023}} {{pb}}

{{cite journal |mode=cs2 |last=

Barbier |first=Joseph-Émile |author-link=Joseph-Émile Barbier |year=

1864 |title=Démonstration du théorème de Lexell |trans-title=Proof of Lexell's theorem |language=fr |journal=Les Mondes |volume=4 |pages=42–43 |url=https://books.google.com/books?id=dsgWAQAAIAAJ&pg=PA42 }} {{pb}}

{{wikicite |ref = {{harvid|Fejes Tóth|1953}}

|reference = {{cite book |mode=cs2 |ref=none |last=

Fejes Tóth |first=László |author-link=László Fejes Tóth |year=

1953 |chapter=§ 1.8 Polare Dreiecke, der Lexellsche Kreis |title=Lagerungen in der Ebene auf der Kugel und in Raum |language=de |series=Die Grundlehren der mathematischen Wissenschaften |volume=65 |publisher=Springer |chapter-url=https://archive.org/details/lagerungenindere0000feje/page/22/ |chapter-url-access=limited |pages=22–23

}}, 2nd ed. 1972, {{doi|10.1007/978-3-642-65234-9_1}}, translated as {{cite book |mode=cs2 |last=

Fejes Tóth |first=László |display-authors=0 |translator1-last=

Fejes Tóth |translator1-first=Gábor |translator2-last=

Kuperberg |translator2-first=Włodzimierz |year=

2023 |title=Lagerungen: Arrangements in the Plane, on the Sphere, and in Space |chapter=§ 1.8 Polar Triangles, Lexell's Circle |pages=25–26 |doi=10.1007/978-3-031-21800-2_1 }}

}} {{pb}}

The polar dual to Lexell's theorem had been previously proved trigonometrically by A. N. J. Sorlin (1825); see #Sorlin's theorem (polar dual) below.

{{cite journal |mode=cs2 |last1=

Praun |first1=Emil |last2=

Hoppe |first2=Hugues |year=

2003 |title=Spherical parametrization and remeshing |journal=ACM Transactions on Graphics |volume=22 |number=3 |pages=340–349 |url=http://hhoppe.com/sphereparam.pdf |doi=10.1145/882262.882274 }} {{pb}}

{{cite journal |mode=cs2 |last=

Carfora |first=Maria Francesca |year=

2007 |title=Interpolation on spherical geodesic grids: A comparative study |journal=Journal of Computational and Applied Mathematics |volume=210 |number=1–2 |pages=99–105 |doi=10.1016/j.cam.2006.10.068 |doi-access=free }} {{pb}}

{{cite journal |mode=cs2 |last1=

Lei |first1=Kin |last2=

Qi |first2=Dongxu |last3=

Tian |first3=Xiaolin |title=A new coordinate system for constructing spherical grid systems |journal=Applied Sciences |volume=10 |number=2 |year=2020 |pages=655 |doi=10.3390/app10020655 |doi-access=free }}

{{harvnb|Atzema|2017}}, {{harvnb|Maehara|Martini|2023}} {{pb}}

Gauss wrote this proof in a letter to Heinrich Christian Schumacher in 1841, in response to a related proof from Thomas Clausen sent to him by Schumacher. The correspondence was later published in: {{pb}}

{{cite book |mode=cs2 |author1-last=

Gauss |author1-first=Carl Friedrich |author1-link=Carl Friedrich Gauss |author2-last=

Schumacher |author2-first=Heinrich Christian |author2-link=Heinrich Christian Schumacher |year=

1862 |editor-last=Peters |editor-first=Christian August Friedrich |editor-link=Christian August Friedrich Peters |title=Briefwechsel zwischen C. F. Gauss und H. C. Schumacher |volume=4 |publisher=Gustav Esch |pages=46–49 |url=https://archive.org/details/bub_gb_wkYvBhl9XaQC/page/n59/ }} {{pb}}

The same proof can also be found in: {{pb}}

{{cite journal |mode=cs2 |last=

Persson |first=Ulf |year=

2012 |title=Lexell's Theorem |journal=Normat |volume=60 |number=3 |pages=133–134 |url=https://normat.ncm.gu.se/pdf_html/2012-03-133.pdf }}

{{harvnb|Maehara|Martini|2023}} {{pb}}

{{cite book |mode=cs2 |last=

Serret |first=Paul |author-link=Paul Serret |year=

1855 |title=Des méthodes en géométrie |trans-title=Methods in geometry |language=fr |chapter=§ 2.3.24 Démonstration du théorème de Lexell. – Énoncé d'un théorème de M. Steiner. – Construction du demi-excès sphérique. |trans-chapter=Proof of Lexell's theorem. – Statement of a theorem of Mr. Steiner. – Construction of the spherical half-excess. |publisher=Mallet-Bachelier |pages=31–34 |chapter-url=https://archive.org/details/desmthodeseng00serruoft/page/31/ }} {{pb}}

{{cite journal |mode=cs2 |last=

Simonič |first=Aleksander |year=

2019 |title=Lexell's theorem via stereographic projection |journal=Beiträge zur Algebra und Geometrie |volume=60 |number=3 |pages=459–463 |doi=10.1007/s13366-018-0426-2 }} {{pb}}

{{cite journal |mode=cs2 |last1=

Maehara |first1=Hiroshi |last2=

Martini |first2=Horst |year=

2022 |title=On Cesàro triangles and spherical polygons |journal=Aequationes Mathematicae |volume=96 |number=2 |pages=361–379 |doi=10.1007/s00010-021-00820-y }}

{{harvnb|Todhunter|Leathem|1901}}, [https://archive.org/details/sphericaltrigono00todh/page/118/ § 153. Lexell's locus], {{pgs|118–119}}

{{harvnb|Lexell|1784}};

{{harvnb|Euler|1797}};

{{harvnb|Casey|1889}}, [https://archive.org/details/treatiseonspheri00seri/page/n108/ 5.2 Lexell's Theorem], {{nobr|§§ 88–91}}, {{pgs|92–97}};

{{harvnb|Todhunter|Leathem|1901}}, [https://archive.org/details/sphericaltrigono00todh/page/118/ § 153. Lexell's locus], {{pgs|118–119}};

{{harvnb|Maehara|Martini|2023}} {{pb}}

{{cite book |mode=cs2 |last=

Legendre |first=Adrien-Marie |author-link=Adrien-Marie Legendre |year=

1800 |title=Éléments de géométrie, avec des notes |trans-title=Elements of geometry, with notes |language=fr |edition=3rd |publisher=Firmin Didot |chapter=Note X, Problème III. Déterminer sur la surface de la sphère la ligne sur laquelle sont situés tous les sommets des triangles de même base et de même surface. |trans-chapter=Determine on the surface of the sphere the curve on which are located all the vertices of the triangles with the same base and the same surface area |pages=320–321 in the 15th edition (1862, for which a better scan is available), [https://archive.org/details/lmentsdego00lege/page/n467/mode/1up figure 285 pl. 13] |chapter-url=https://archive.org/details/lmentsdego00lege/page/320/ }} {{pb}}

{{cite book |mode=cs2 |last=

Puissant |first=Louis |author-link=Louis Puissant |year=

1842 |title=Traité de géodésie |trans-title=Treatise on Geodesy |language=fr |volume=1 |edition=3rd |publisher=Bachelier |pages=114–115 |url=https://babel.hathitrust.org/cgi/pt?id=wu.89078557931&seq=136 }} {{pb}}

{{cite book |mode=cs2 |last=

Le Cointe |first=Ignace-Louis-Alfred |author-link=Ignace-Louis-Alfred Le Cointe |year=

1858 |title=Leçons sur la théorie des fonctions circulaires et la trigonométrie |language=fr |chapter=Théorème de Lexell |trans-title=Lessons on the theory of circular functions and trigonometry |publisher=Mallet-Bachelier |chapter-url=https://babel.hathitrust.org/cgi/pt?id=ucm.5305751310&seq=279 |at={{nobr|§§ 181–182}}, {{pgs|263–265}} }} {{pb}}

{{cite book |mode=cs2 |last=

Serret |first=Joseph-Alfred |author-link=Joseph-Alfred Serret |year=

1862 |title=Traité de trigonométrie |trans-title=Treatise on trigonometry |edition=3rd |publisher=Mallet-Bachelier |chapter=Expressions du rayon du cercle circonscrit et des rayons des cercles inscrit et exinscrits. |trans-chapter=Expressions of the radius of the circumscribed circle and the radii of the inscribed and exscribed circles. |language=fr |at={{nobr|§ 94}}, {{pgs|141–142}} |chapter-url=https://babel.hathitrust.org/cgi/pt?id=chi.11707201&seq=157 }}

}}

{{refbegin|30em|indent=yes}}

• {{cite conference |mode=cs2 |last=

Atzema |first=Eisso J. |year=

2017 |contribution = 'A Most Elegant Property': On the Early History of Lexell’s Theorem |conference=CSHPM 2016, Calgary, Alberta |title=Research in History and Philosophy of Mathematics |editor1-last=Zack |editor1-first=Maria |editor2-last=Schlimm |editor2-first=Dirk |pages=117–132 |publisher=Birkhäuser |doi=10.1007/978-3-319-64551-3_8 }}

• {{cite book |mode=cs2 |last=

Casey |first=John |author-link=John Casey (mathematician) |year=

1889 |title=A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples |location=Dublin |publisher=Hodges, Figgis, & Co. |url=https://archive.org/details/treatiseonspheri00seri/ }}

• {{wikicite |ref = {{harvid|Euler|1797}}

|reference = {{cite journal |mode=cs2 |ref=none |last=

Euler |first=Leonhard |author1-link=Leonhard Euler |year=

1797 |orig-year=written 1778 |title=Variae speculationes super area triangulorum sphaericorum |journal=Nova Acta Academiae Scientiarum Imperialis Petropolitanae |volume=10 |pages=47–62; [https://archive.org/details/novaactaacademia10petr/page/n781/mode/1up figures tab. 1] |id=E[https://scholarlycommons.pacific.edu/euler-works/698/ 698] |language=la |url=https://archive.org/details/novaactaacademia10petr/page/47/

}}, in Opera omnia, ser. 1, vol. 29, {{pgs|253–266}}, translated as {{cite web |mode=cs2 |last=

Euler |first=Leonhard |display-authors=0 |translator-last=

Stén |translator-first=Johan Carl-Erik |year=

2008 |work=17centurymaths.com |title=Different Speculations on the Area of Spherical Triangles |url=https://17centurymaths.com/contents/euler/e698tr.pdf }}

}}

• {{wikicite |ref = {{harvid|Lexell|1784}}

|reference = {{cite journal |mode=cs2 |ref=none |last=

Lexell |first=Anders Johan |author-link=Anders Johan Lexell |year=

1784 |orig-date=written c. 1777 |title=Solutio problematis geometrici ex doctrina sphaericorum |language=la |journal=Acta Academiae Scientarum Imperialis Petropolitinae |volume=5: 1781 |number=1 |pages=112–126; [https://archive.org/details/actaacademiaesci81impe/page/n473/mode/1up figures tab. 4] |url=https://archive.org/details/actaacademiaesci81impe/page/112/

}}, translated as {{cite web |mode=cs2 |last=

Lexell |first=Anders Johan |display-authors=0 |translator-last=

Stén |translator-first=Johan Carl-Erik |year=

2009 |work=17centurymaths.com |title=Solution of a geometrical problem from the theory of the sphere |url=https://17centurymaths.com/contents/euler/lexellone.pdf }}

}}

• {{cite journal |mode=cs2 |last1=

Maehara |first1=Hiroshi |last2=

Martini |first2=Horst |year=

2023 |title=Seven Proofs of Lexell's Theorem: An Excursion into Spherical Geometry |journal=Mathematical Intelligencer |doi=10.1007/s00283-023-10281-7 }}

• {{cite journal |mode=cs2 |last=

Papadopoulos |first=Athanase |author-link=Athanase Papadopoulos |year=

2014 |title=On the works of Euler and his followers on spherical geometry |journal=Gaṇita Bhārati |volume=36 |pages=53–108 |arxiv=1409.4736 }}

• {{cite book |mode=cs2 |last=

Stén |first=Johan Carl-Erik |year=

2014 |title=A Comet of the Enlightenment: Anders Johan Lexell's Life and Discoveries |publisher=Birkhäuser |doi=10.1007/978-3-319-00618-5 }}

• {{cite book |mode=cs2 |last1=

Todhunter |first1=Isaac |author1-link=Isaac Todhunter |last2=

Leathem |first2=John Gaston |year=

1901 |title=Spherical Trigonometry |edition=Revised |publisher=MacMillan |url=https://archive.org/details/sphericaltrigono00todh/ }}

• {{cite journal |mode=cs2 |last=

Zhukova |first=Alena M. |year=

2019 |title=On the Contribution of Anders Johan Lexell in Spherical Geometry |journal=Gaṇita Bhārati |volume=41 |number=1–2 |pages=127–149 |doi=10.32381/GB.2019.41.1-2.5 |id={{ProQuest|2561520777}} }}

{{refend}}

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