Part 5 of sharing Catriona Shearer’s geometry problems with my younger son

Yesterday we looked at this interesting geometry puzzle from Catriona Shearer:

What fraction of this equilateral triangle is shaded? pic.twitter.com/quhp02AM3Z

— Catriona Shearer (@Cshearer41) April 10, 2020

My son took a pretty computational approach to the problem and the calculations gave him a little difficulty. Today my main idea was to share Ian Agol’s neat solution, but I wanted to review one piece of yesterday’s calculation with him as an arithmetic review:

Now we moved on to looking at Ian Agol’s solution (using the picture made by Greg Egan):

1/3. Rotate this picture around the bottom vertex v to get 12 tangent circles equidistant to v. Since the circle picture is symmetric, we can rotate by π/6 about v, preserving the circle configuration. This sends the aqua hexagon to one whose vertices lie at the centers.

— Ian Agol (@agolian) April 10, 2020

pic.twitter.com/uS0hBavBW6

— Greg Egan (@gregeganSF) April 10, 2020

My son studied this solution and picture for about 10 min before we started the project. Here’s how he explained it:

I liked following up yesterday’s computational solution with this terrific non-computational one.