To prove the Butterfly Theorem we must first learn how to construct the theorem.
Start by constructing a circle.

Butterfly Theorem

Next, we need to create a chord or diameter of the circle. We will label this PQ

Now, lets fine the midpoint of PQ and label it point M
We will also at this time construct two more chords that will go through our point M.
We will label these AB and CD

Lastly, we will construct two more chords connecting our points AD and CB, that will intersect our chord PQ.
At this time we can also label our exterior angles created by PA and QC, X and Y correspondingly.