Mar 6, 2014
Mar 1, 2014
The Butterfly Theorem

Fig. 2, the butterfly in the butterfly theorem 
Fig. 3, proof of the theorem. O is the center of the circle. M, N are midpoint of the chord PV and UQ respectively. 
Poof:
Let M, N be the midpoint of chord PV and UQ respectively. O is the center of the circle.Points P, V, Q, U are on the same circle. So $\angle VPQ = \angle VUQ$ and $\angle PVU = \angle PQU$. So $\triangle CPV \simeq \triangle CUQ$. So $\frac{MV}{NQ} = \frac{PV/2}{UQ/2} = \frac{PV}{UQ} = \frac{VC}{QC}$.
$\frac{MV}{NQ} = \frac{VC}{QC}$ plus $\angle PVU = \angle PQU$ implies $\triangle MVC \simeq \triangle CQN$. Thus $\angle VMC = \angle CNQ$.
OM is perpendicular to PV. ON is perpendicular to UQ. OC is perpendicular to EF (CX, CY). So O, M, X, C are on the same circle. O, C, Y, N are on the same circle. Thus, $\angle XOC = \angle XMC = \angle YNC = \angle YOC$. Note that OC is perpendicular to EF. Therefore, CX = CY.
The proof of the theorem gives us another butterfly (Fig. 4), which also consists of a pair of similar triangles.
Fig. 4. 
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