This morning I picked what I thought was a medium level difficulty problem for a movie with my older son. The problem was based on the first challenge problem in the back of the chapter on triangles in our *Introduction to Geometry* book. Here’s the problem:

“The angle bisector from one vertex of a triangle passes through the center of the circumscribed circle. Prove that the triangle is isosceles.”

By no means a simple proof, but I thought it would lead to a good conversation. Instead he got stuck and I didn’t do a great job helping. The entire 13 minute process is, I hope, a good problem solving lesson, though. Learning math isn’t always a straight line.

Part 1: Introduction to the problem (about 4 min)

The three things that happen in this video are:

(1) We talk through the problem,

(2) We talk a little bit about properties of perpendicular bisectors, and

(3) We eventually arrive at a reasonable picture for the problem

So, a pretty good start to the problem:

Part 2: The beginning of the proof (about 4 min)

There’s a tiny bit of overlap with the last video. I cut these videos by time rather than by content.

The four things that happen in this video are:

(1) He realizes that the perpendiculars from the circumcenter to the sides are equal,

(2) He has a good guess for which two sides are likely to be equal,

(3) He gets a little confused about the picture. In the picture that we’ve drawn, one of the perpendicular bisectors looks similar to the angle bisector and that causes some confusion, and

(4) He identifies two triangles that are congruent

So, it turns out that the last movie ends in a spot that is just one step away from the final proof. That’s good. But that last step turns out to be a little bit elusive. Part of the difficulty is that in getting to this point my son has lost some of the information that is important from the beginning of the problem.

That happens all the time in problem solving, and is why I think this exercise is a good example. I was reminded of a similar thing happening to Tim Gowers when he live blogged a solution to an International Mathematics Olympiad problem. I wrote a bit about Gowers’s live blogging here:

Problem Solving and Tim Gowers’s live blogging an IMO problem

For the point I’m making here, the important line from Gowers’s blog is this self-deprecating one: “You idiot Gowers, read the question: the a_n have to be positive integers.”

When all problem solving in math is presented as an easy straight line process, you forget that everyone from kids to Fields medal winners sometimes lose the thread of a problem they are working on.

Part 3: (about 5 min)

Here is where I struggle. We are so close to the solution which made me reluctant to point him in the right direction. Unfortunately we were also up against a time deadline since our neighborhood dog walk happens at 7:00 am and going on that walk is important to my kids. May seem silly, but the time pressure was real to my older son.

In this video:

(1) I try to help him see that the perpendiculars also bisect the sides, but that’s the point that he’s forgotten, unfortunately

(2) I have him go back and read the problem again,

(3) Once he realizes that the circumcenter is the intersection of the perpendicular bisectors, he finds the thread of the problem again and we finish the proof.

Part 4: (about 5 min)

After he returned from the dog walk I thought it would help him get a better understanding of the proof if we went through it again. This was just to help him learn, but I thought it would also be good to film it to show what a totally misleading presentation of what learning math looks like. Everything goes perfectly smoothly in this one!

## Comments

I liked the problem, and when I drew a diagram it started with the circumcircle and three points on it. Taking the top one as the vertex of interest and joining the three vertices to the circumcentre I saw two iscoceles triangles. The angles at the top are equal (given) so the two triangles are congruent. Done !

Not such a hard problem when you see the circumcircle.

Clever solution