In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.
Let
O
R
O1,O2,O3,O4
O
tij
Oi,Oj
t12 ⋅ t34+t14 ⋅ t23=t13 ⋅ t24.
Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.
The following proof is attributable to Zacharias. Denote the radius of circle
Oi
Ri
O
Ki
O,Oi
2=\overline{O | |
t | |
iO |
2-(R | |
i-R |
2. | |
j) |
We will try to express this length in terms of the points
Ki,Kj
OiOOj
\overline{OiO
2-2\overline{OO | |
i} ⋅ |
\overline{OOj} ⋅ \cos\angleOiOOj
Since the circles
O,Oi
\overline{OOi}=R-Ri,\angleOiOOj=\angleKiOKj
Let
C
O
KiCKj
\overline{KiKj}=2R ⋅ \sin\angleKiCKj=2R ⋅ \sin
\angleKiOKj | |
2 |
Therefore,
\cos\angleKiOKj=
| ||||
1-2\sin |
=1-2 ⋅ \left(
\overline{KiKj | |
and substituting these in the formula above:
\overline{OiO
2-2(R-R | |
i)(R-R |
2}\right) | ||||
|
\overline{OiO
2-2(R-R | |
i)(R-R |
j)+(R-Ri)(R-Rj) ⋅
\overline{KiKj | |
2}{R |
2}
\overline{OiO
2=((R-R | |
i)-(R-R |
2+(R-R | |
i)(R-R |
j) ⋅
\overline{KiKj | |
2}{R |
2}
And finally, the length we seek is
tij=\sqrt{\overline{OiO
2-(R | |
i-R |
| ||||
j) |
\sqrt{R-Rj} ⋅ \overline{KiKj}}{R}
K1K2K3K4
\begin{align} &t12t34+t14t23\\[4pt] ={}&
1 | |
R2 |
⋅ \sqrt{R-R1}\sqrt{R-R2}\sqrt{R-R3}\sqrt{R-R4}\left(\overline{K1K2} ⋅ \overline{K3K4}+\overline{K1K4} ⋅ \overline{K2K3}\right)\\[4pt] ={}&
1 | |
R2 |
⋅ \sqrt{R-R1}\sqrt{R-R2}\sqrt{R-R3}\sqrt{R-R4}\left(\overline{K1K3} ⋅ \overline{K2K4}\right)\\[4pt] ={}&t13t24\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
Oi,Oj
O
tij
If
Oi,Oj
O
tij
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 of Feuerbach's theorem uses the converse theorem.