The kids got back from an overnight camp yesterday and were a little tired. I figured the energy would be a little low and wanted to pick a problem that would would be instructive as well as challenging. This problem that my older son adn I talked about last week seemed like a good fit:
Since my older son had already going through the problem, I tried to ask my younger son most of the questions in this project and my older son come in when my younger son was stuck.
One small note – I’m writing this post at my son’s archery club and I have to have the sound off for the videos (I forgot my headphones) so the descriptions of the videos is going to be a little lighter than usual.
Here’s the introduction to the problem:
The first idea we talked about was finding the center of the inscribed circle in a right triangle. There are a couple of directions we could have gone here. I thought about discussing the 
Next we talked about some special properties about the circumscribed circle in a right triangle. The geometric idea here is that a triangle inscribed in a semi circle is a right triangle.
Finally, now that we’d found both the incenter and circumcenter of the triangle, we talked about how to find the distance between those two points. my younger son got confused by a small arithmetic point.
So despite the low energy, a fun project. I like using the old AMC problems to motive some basic math ideas. It is also fun to see the difference between how the boys react to seeing ideas for the first time vs the 10th time.
Mike's Math Page 
I didn’t watch these carefully, but in skim-skipping through your part C, I notice you didn’t draw what is for me the key diagram explaining Thales’s (probably misattributed) semicircle theorem: you can duplicate and rotate your right triangle half a turn and glue it back to itself to get a rectangle, and of course the two diagonals of a rectangle must have the same length, and therefore the four lengths from the center of the rectangle to the corners must all be the same.
This little fact about circles is of great use in practical craft-type applications (drawing, carpentry, …), for example finding the diameter of a given circle through a particular point, finding the center of some existing circular object (or a circle traced from one), accurately drawing the tangent to a circle, etc.