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In [[mathematics]], '''Casey's theorem''', also known as the generalized [[Ptolemy's theorem]], is a theorem in [[Euclidean geometry]] named after the Irish [[mathematics|mathematician]] [[John Casey (mathematician)|John Casey]].
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== Formulation of the theorem==
 
[[Image:Theorem of casey2.png|thumb|350px|<math>t_{12} \cdot t_{34}+t_{41}\cdot t_{23}-t_{13}\cdot t_{24}=0 </math>]]
 
Let <math>\,O</math> be a circle of radius <math>\,R</math>. Let <math>\,O_1, O_2, O_3, O_4</math> be (in that order) four non-intersecting circles that lie inside <math>\,O</math> and tangent to it. Denote by <math>\,t_{ij}</math> the length of the exterior common tangent of the circles <math>\,O_i, O_j</math>. Then:<ref name="Cas"/>
 
:<math>\,t_{12} \cdot t_{34}+t_{41} \cdot t_{23}=t_{13}\cdot t_{24}.</math>
 
Note that in the degenerate case, where all four circles reduce to points, this is exactly [[Ptolemy's theorem]].
 
== Proof ==
 
The following proof is due<ref name="Bot"/> to Zacharias.<ref name="Zach"/> Denote the radius of circle <math>\,O_i</math> by <math>\,R_i</math> and its tangency point with the circle <math>\,O</math> by <math>\,K_i</math>. We will use the notation <math>\,O, O_i</math> for the centers of the circles.
Note that from [[Pythagorean theorem]],
 
:<math>\,t_{ij}^2=\overline{O_iO_j}^2-(R_i-R_j)^2.</math>
 
We will try to express this length in terms of the points <math>\,K_i,K_j</math>. By the [[law of cosines]] in triangle <math>\,O_iOO_j</math>,
 
:<math>\overline{O_iO_j}^2=\overline{OO_i}^2+\overline{OO_j}^2-2\overline{OO_i}\cdot \overline{OO_j}\cdot \cos\angle O_iOO_j</math>
 
Since the circles <math>\,O,O_i</math> tangent to each other:
 
:<math>\overline{OO_i} = R - R_i,\, \angle O_iOO_j = \angle K_iOK_j</math>
 
Let <math>\,C</math> be a point on the circle <math>\,O</math>. According to the [[law of sines]] in triangle <math>\,K_iCK_j</math>:
 
:<math>\overline{K_iK_j} = 2R\cdot \sin\angle K_iCK_j = 2R\cdot \sin\frac{\angle K_iOK_j}{2}</math>
 
Therefore,
 
:<math>\cos\angle K_iOK_j = 1-2\sin^2\frac{\angle K_iOK_j}{2}=1-2\cdot \left(\frac{\overline{K_iK_j}}{2R}\right)^2 = 1 - \frac{\overline{K_iK_j}^2}{2R^2}</math>
 
and substituting these in the formula above:
 
:<math>\overline{O_iO_j}^2=(R-R_i)^2+(R-R_j)^2-2(R-R_i)(R-R_j)\left(1-\frac{\overline{K_iK_j}^2}{2R^2}\right)</math>
 
:<math>\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)\cdot \frac{\overline{K_iK_j}^2}{R^2}</math>
 
:<math>\overline{O_iO_j}^2=((R-R_i)-(R-R_j))^2+(R-R_i)(R-R_j)\cdot \frac{\overline{K_iK_j}^2}{R^2}</math>
 
And finally, the length we seek is
 
:<math>t_{ij}=\sqrt{\overline{O_iO_j}^2-(R_i-R_j)^2}=\frac{\sqrt{R-R_i}\cdot \sqrt{R-R_j}\cdot \overline{K_iK_j}}{R}</math>
 
We can now evaluate the left hand side, with the help of the original [[Ptolemy's theorem]] applied to the inscribed [[quadrilateral]] <math>\,K_1K_2K_3K_4</math>:
 
:<math>t_{12}t_{34}+t_{14}t_{23}=\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)</math>
 
:<math>=\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)=t_{13}t_{24}</math>
 
Q.E.D.
 
==Further generalizations ==
 
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:<ref name="John"/>
:
If <math>\,O_i, O_j</math> are both tangent from the same side of <math>\,O</math> (both in or both out), <math>\,t_{ij}</math> is the length of the exterior common tangent.
:
If <math>\,O_i, O_j</math> are tangent from different sides of <math>\,O</math> (one in and one out), <math>\,t_{ij}</math> is the length of the interior common tangent.
:
It is also worth noting that the converse of this statement is also true.<ref name="John"/> That is, if equality holds, the circles are tangent.
 
==Applications==
 
Casey's theorem and its converse can be used to prove a variety of statements in [[Euclidean geometry]].  
:
For example, the shortest known proof<ref name="Cas"/> of [[Feuerbach's theorem]] uses the converse theorem.
 
==References==
 
<references>
 
<ref name="Cas">
{{cite journal
  | first = J.
  | author = Casey
  | journal = Math. Proc. R. Ir. Acad.
  | volume = 9
  | pages = 396
  | year = 1866
  |separator = ,
}}
</ref>
 
<ref name="Zach">
{{cite journal
  | first= M.
  | last = Zacharias
  | journal = [[Jahresbericht der Deutschen Mathematiker-Vereinigung]]
  | volume = 52
  | year = 1942
  | separator = ,
}}
</ref>
 
<ref name="Bot">
{{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
  | separator = ,
}}
</ref>
 
<ref name="John">
{{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
  | separator = ,
}}
 
</ref>
 
</references>
 
==External links==
* {{MathWorld|urlname=CaseysTheorem|title=Casey's theorem}}
* [http://journals.cms.math.ca/cgi-bin/vault/public/view/CRUXv22n2/body/PDF/page49-53.pdf?file=page49-53 Shailesh Shirali: ''On a generalized Ptolemy Theorem'']
 
[[Category:Circles]]
[[Category:Euclidean geometry]]
[[Category:Theorems in geometry]]
[[Category:Articles containing proofs]]

Revision as of 20:44, 15 February 2014

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