The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:
A formal proof of the theorem is as follows: Let the perpendiculars and be dropped from the point on the straight lines and respectively. Similarly, let and be dropped from the point perpendicular to the straight lines and respectively.
From the preceding equations, it can be easily seen that
So, it can be concluded that or is the midpoint of
An alternate proof using projective geometry can be found in problem 8 of the link below.
H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967.