One part of math that I love is the sense of astonishment when you discover something new in math. This “ah-ha” moment is why I love math so much. It drives my passion for math, and getting better at it. An “ah-ha” moment that I want to share with you is one relating to geometry. There are a lot of astonishing proofs in geometry, but this is just one. It is so simple and elegant, and understandable by anyone.

Here is the statement: Any angle inscribed in a semicircle (the two outer vertices are touching the corners of the semicircle, and the middle touches the curve) is a right angle! The illustration below will help visualize.

The proof for this statement is surprisingly simple. The next image explains it well.

If you were to reflect a semicircle (in this case across AB) then the ensuing shape will be a rectangle. The image above shows that happening. So no matter how you inscribe the angle (look it up if you still don’t get inscribed angles) the resulting angle will always be a right angle if you drew it in a semicircle.

How cool is that?