Casey's theorem

File:Casey new1a.svg

Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.

:$\backslash ,t\_\{12\}\; \backslash cdot\; t\_\{34\}+t\_\{14\}\; \backslash cdot\; t\_\{23\}=t\_\{13\}\backslash cdot\; t\_\{24\}.$

The following proof is attributable to Zacharias. Denote the radius of circle $\backslash ,O\_i$ by $\backslash ,R\_i$ and its tangency point with the circle $\backslash ,O$ by $\backslash ,K\_i$. We will use the notation $\backslash ,O,\; O\_i$ for the centers of the circles.

Note that from Pythagorean theorem,

:$\backslash ,t\_\{ij\}^2=\backslash overline\{O\_iO\_j\}^2-(R\_i-R\_j)^2.$

:$\backslash overline\{O\_iO\_j\}^2=\backslash overline\{OO\_i\}^2+\backslash overline\{OO\_j\}^2-2\backslash overline\{OO\_i\}\backslash cdot\; \backslash overline\{OO\_j\}\backslash cdot\; \backslash cos\backslash angle\; O\_iOO\_j$

Since the circles $\backslash ,O,O\_i$ tangent to each other:

:$\backslash overline\{OO\_i\}\; =\; R\; -\; R\_i,\backslash ,\; \backslash angle\; O\_iOO\_j\; =\; \backslash angle\; K\_iOK\_j$

Let $\backslash ,C$ be a point on the circle $\backslash ,O$. According to the law of sines in triangle $\backslash ,K\_iCK\_j$:

:$\backslash overline\{K\_iK\_j\}\; =\; 2R\backslash cdot\; \backslash sin\backslash angle\; K\_iCK\_j\; =\; 2R\backslash cdot\; \backslash sin\backslash frac\{\backslash angle\; K\_iOK\_j\}\{2\}$

Therefore,

:$\backslash cos\backslash angle\; K\_iOK\_j\; =\; 1-2\backslash sin^2\backslash frac\{\backslash angle\; K\_iOK\_j\}\{2\}=1-2\backslash cdot\; \backslash left(\backslash frac\{\backslash overline\{K\_iK\_j\}\}\{2R\}\backslash right)^2\; =\; 1\; -\; \backslash frac\{\backslash overline\{K\_iK\_j\}^2\}\{2R^2\}$

and substituting these in the formula above:

:$\backslash overline\{O\_iO\_j\}^2=(R-R\_i)^2+(R-R\_j)^2-2(R-R\_i)(R-R\_j)\backslash left(1-\backslash frac\{\backslash overline\{K\_iK\_j\}^2\}\{2R^2\}\backslash right)$

:$\backslash overline\{O\_iO\_j\}^2=(R-R\_i)^2+(R-R\_j)^2-2(R-R\_i)(R-R\_j)+(R-R\_i)(R-R\_j)\backslash cdot\; \backslash frac\{\backslash overline\{K\_iK\_j\}^2\}\{R^2\}$

:$\backslash overline\{O\_iO\_j\}^2=((R-R\_i)-(R-R\_j))^2+(R-R\_i)(R-R\_j)\backslash cdot\; \backslash frac\{\backslash overline\{K\_iK\_j\}^2\}\{R^2\}$

And finally, the length we seek is

:$t\_\{ij\}=\backslash sqrt\{\backslash overline\{O\_iO\_j\}^2-(R\_i-R\_j)^2\}=\backslash frac\{\backslash sqrt\{R-R\_i\}\backslash cdot\; \backslash sqrt\{R-R\_j\}\backslash cdot\; \backslash overline\{K\_iK\_j\}\}\{R\}$

We can now evaluate the left hand side, with the help of the original Ptolemy's theorem applied to the inscribed quadrilateral $\backslash ,K\_1K\_2K\_3K\_4$:

:$$

\begin{align}

& t_{12}t_{34}+t_{14}t_{23} \\[4pt]

= {} & \frac{1}{R^2}\cdot \sqrt{R-R_1}\sqrt{R-R_2}\sqrt{R-R_3}\sqrt{R-R_4} \left(\overline{K_1K_2} \cdot \overline{K_3K_4}+\overline{K_1K_4}\cdot \overline{K_2K_3}\right) \\[4pt]

= {} & \frac{1}{R^2}\cdot \sqrt{R-R_1}\sqrt{R-R_2}\sqrt{R-R_3}\sqrt{R-R_4}\left(\overline{K_1K_3}\cdot \overline{K_2K_4}\right) \\[4pt]

= {} & t_{13}t_{24}

\end{align}

It can be seen that the four circles need not lie inside the big circle. In fact, they may be tangent to it from the outside as well. In that case, the following change should be made:

:

If $\backslash ,O\_i,\; O\_j$ are both tangent from the same side of $\backslash ,O$ (both in or both out), $\backslash ,t\_\{ij\}$ is the length of the exterior common tangent.

:

If $\backslash ,O\_i,\; O\_j$ are tangent from different sides of $\backslash ,O$ (one in and one out), $\backslash ,t\_\{ij\}$ is the length of the interior common tangent.

:

The converse of Casey's theorem is also true. That is, if equality holds, the circles are tangent to a common circle.

Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof{{rp|411}} of Feuerbach's theorem uses the converse theorem.

{{cite journal

| last = Casey | first = J. | author-link = John Casey (mathematician)

| journal = Proceedings of the Royal Irish Academy

| jstor = 20488927

| pages = 396–423

| title = On the Equations and Properties: (1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane

| volume = 9

| year = 1866}}

{{cite journal

| first= M.

| last = Zacharias

| journal = Jahresbericht der Deutschen Mathematiker-Vereinigung

| volume = 52

| year = 1942

|pages=79–89

|title=Der Caseysche Satz}}

{{cite book

| first = O.

| last = Bottema

| title = Hoofdstukken uit de Elementaire Meetkunde

| publisher = (translation by Reinie Erné as Topics in Elementary Geometry, Springer 2008, of the second extended edition published by Epsilon-Uitgaven 1987)

| year = 1944}}

{{cite book

| first = Roger A.

| last = Johnson

| title = Modern Geometry

| publisher = Houghton Mifflin, Boston (republished facsimile by Dover 1960, 2007 as Advanced Euclidean Geometry)

| year = 1929}}

{{Commons category|Casey's theorem}}

• {{MathWorld|urlname=CaseysTheorem|title=Casey's theorem}}
• Shailesh Shirali: [https://cms.math.ca/publications/crux/issue/?volume=22&issue=2 "'On a generalized Ptolemy Theorem'"]. In: Crux Mathematicorum, Vol. 22, No. 2, pp. 49-53

Category:Theorems about circles

Category:Euclidean geometry

Category:Articles containing proofs