The butterfly theorem is a well-known result from Euclidean geometry.
Let be the midpoint of chord of a circle.
Draw two other chords and through .
If and intersect chord at and , respectively, is the midpoint of .
Looking at the diagram, you can probably tell how the butterfly theorem got its name!
There are various proofs for the butterfly theorem. We're going to prove it by using the properties of similar triangles.
We're going to start by dropping perpendiculars and from to and , respectively, and likewise and from to and , respectively.
Here's what the diagram looks like after we do that:
Let , , and .
Now we have a lot of similar triangles we can work with, and from these similar triangles we have a lot of ratios that are equal to one another:
So, we have .
Now we're going to use the intersecting chords theorem. (Check the wiki out if you aren't familiar with it.)
From that theorem, we have, .
So, we can write .
In other words, . But that can only happen if .
This means , which implies is the midpoint of .
- Let the incircle with center of touch the side at , and let be the midpoint of this side. Then prove that line (extended) bisects .
- Let and be two tangents to a circle, the diameter through , and the perpendicular from to . Then prove that bisects .