Lemoine point

{{Short description|Intersection of the three symmedian lines of a triangle}}

{{distinguish|Lemoine Point Conservation Area}}

File:Lemoine punkt.svg (black), angle bisectors (dotted) and symmedians (red). The symmedians intersect in the symmedian point L, the angle bisectors in the incenter I and the medians in the centroid G.]]

In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle. In other words, it is the isogonal conjugate of the centroid.

Ross Honsberger called its existence "one of the crown jewels of modern geometry".

In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6).[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers], accessed 2014-11-06. For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.{{citation|first1=Christopher J.|last1=Bradley|first2=Geoff C.|last2=Smith|title=The locations of triangle centers|journal=Forum Geometricorum|volume=6|year=2006|pages=57–70|url=http://forumgeom.fau.edu/FG2006volume6/FG200607index.html|access-date=2016-10-18|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304000426/http://forumgeom.fau.edu/FG2006volume6/FG200607index.html|url-status=dead}}.

The symmedian point of a triangle with side lengths {{mvar|a}}, {{mvar|b}} and {{mvar|c}} has homogeneous trilinear coordinates {{math|[a : b : c]}}.

An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the hesse normal forms of the corresponding lines. The solution of this overdetermined system found by the least squares method gives the coordinates of the point. It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides.

The Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle.{{citation

| last1 = Beban-Brkić | first1 = J.

| last2 = Volenec | first2 = V.

| last3 = Kolar-Begović | first3 = Z.

| last4 = Kolar-Šuper | first4 = R.

| journal = Rad Hrvatske Akademije Znanosti i Umjetnosti

| mr = 3100227

| pages = 95–106

| title = On Gergonne point of the triangle in isotropic plane

| volume = 17

| year = 2013}}.

The symmedian point of a triangle {{mvar|ABC}} can be constructed in the following way: let the tangent lines of the circumcircle of {{mvar|ABC}} through {{mvar|B}} and {{mvar|C}} meet at {{mvar|A'}}, and analogously define {{mvar|B'}} and {{mvar|C'}}; then {{mvar|A'B'C'}} is the tangential triangle of {{mvar|ABC}}, and the lines {{mvar|AA'}}, {{mvar|BB'}} and {{mvar|CC'}} intersect at the symmedian point of {{mvar|ABC}}.{{efn|If ABC is a right triangle with right angle at A, this statement needs to be modified by dropping the reference to AA' since the point A' does not exist.}} It can be shown that these three lines meet at a point using Brianchon's theorem. Line {{mvar|AA'}} is a symmedian, as can be seen by drawing the circle with center {{mvar|A'}} through {{mvar|B}} and {{mvar|C}}.{{cn|date=February 2016}}

The French mathematician Émile Lemoine proved the existence of the symmedian point in 1873, and Ernst Wilhelm Grebe published a paper on it in 1847. Simon Antoine Jean L'Huilier had also noted the point in 1809.{{citation|first=Ross|last=Honsberger|authorlink=Ross Honsberger|contribution=Chapter 7: The Symmedian Point|title=Episodes in Nineteenth and Twentieth Century Euclidean Geometry|publisher=Mathematical Association of America|location=Washington, D.C.|year=1995}}.

For the extension to an irregular tetrahedron see symmedian.