
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. 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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