The Power of a Point

My older son and I are studying the Power of a Point chapter in our Introduction to Geometry book right now. Yesterday I saw a fun problem on Art of Problem Solving’s website and thought it would make for a neat discussion with my son.

The problem is Intermediate Problem #1 here:

See Intermediate Problem #1 here

Just after going through the project I saw this tweet from Patrick Honner:

I mentioned to him that we’d just looked at a fun problem which led to this exchange – I hadn’t noticed that the problem could be interpreted in two different ways:

With that introduction, here’s our talk about this problem.

My son is initially surprised that you can even talk about the perimeter of the triangle since you don’t know the lengths of any of the sides. He notices a few other geometric ideas, too, and eventually uses the some of the ideas about the power of a point to conclude / remember that two tangents to a circle from the same point have equal length.

This is the key idea in the problem, though we don’t head directly to the solution after discovering this idea.

So, having walked up to inches away from the solution in the last video, now we finish it. It is pretty surprising that you can say anything about the perimeter of a triangle when you seem to know almost nothing about the triangle. That’s one of the nice surprises of this problem.

After we finish up with his solution, we wrap up by showing another little geometric trick that gives you an idea of what the perimeter might be. Fun little problem: