My younger son has been studying Art of Problem Solving’s Introduction to Geometry book this year. He’s been doing most of the work on his own. Every now and then, though, I decided to check in and see how things are going. Today the topic he was looking at when I checked in was constructions. The two questions on the table when I stopped by were:

(i) Construct the incircle of a triangle, and

(ii) Construct the circumcircle of a triangle

Here’s how things went.

(i) The incircle

He has the basic idea for how the construction works, but misses one important idea. That idea is that the tangents of the incircle are not the feet of the angle bisectors. My guess is that this is a fairly common point of confusion for kids learning this topic:

(ii) the circumcircle

Here his solution was completely correct. He had a tiny bit of trouble in the beginning figuring out to construct the perpendicular bisectors, but he worked through that trouble fairly quickly.

It is always interesting to hear the ideas that kids have when they talk through a mathematical process. For me it was especially nice to hear that most of the ideas he’s learning as he works through this geometry book are sinking in pretty well.