I’ve been studying angles and circles lately with my older son. One of his homework problems from our Introduction to Geometry book was to prove a formula for the radius of the circle inscribed in a right triangle. The problem gave him a bit of difficulty, but talking through it led to a great discussion about tangent lines and angles.

First up, the original problem:

The prior discussion extends quite naturally to give a nice (but different) formula for the radius of the inscribed circle in a general triangle. Really the only difference in the argument is that we are looking at areas this time rather than at lengths:

Finally, we’ve just found two different formulas for the radius of the inscribed circle. For a right triangle, these two formulas ought to be the same. They don’t look the same, though, what’s going on? Oh look, the Pythagorean theorem!

So a fun geometry discussion starting from a nice little homework problem. I remember learning these formulas in high school through prepping for math contests, though I honestly don’t remember how much we dug into the ideas behind the formulas back then. It sure is fun to share these ideas with my son now. It is amazing how some simple ideas about angles in circles lead to such beautiful results.