Mar 1, 2014

The Butterfly Theorem

Fig. 1, C is the midpoint of a chord EF. PQ and UV are two chords passing through C. PV meets EF at X. UQ meets EF at Y. The butterfly theorem says, CX = CY.    
The Butterfly Theorem is a classical result in Euclidean geometry. It gives a beautiful shape of butterfly (Fig. 2) along with two equal length segments.

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. 


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.