Matt Enlow posted an interesting geometry puzzle on twitter (tweet here), and I think the thing that got me intrigued was his initial challenge: “I can’t tell how hard this problem I just made up is.” Not knowing if there is an elegant/easy/obvious solution or not got me hooked.
I’m going to try to outline my approach/solution, because I sometimes like deconstructing my thinking to see how I actually think/learn… so from this point on… SPOILERS.
Some things that stood out to me… First, it looked like there was initially a single circle in a square, and the circle got cut in half and then it started sliding. So I initially drew the full circle in the square (before sliding), I drew the diagram shown, and then I drew the two semicircles in a rectangle after they fully “slid”… I saw the cut circle “in motion” — but after a short while I didn’t see how that would help me.
Then I drew the image and solved the problem and felt proud about it. But then I realized I drew the picture wrong. I circled the wrong part in my diagram, so you can see. I had the “slice” hit the corner of the rectangles, and then I was able to use similar triangles to come up with a solution.
I was proud but for some reason, probably because Matt’s initial tweet suggested to me that it would be harder than this, something was nagging me about it. So I went back and quickly saw my error. But I have always found that taking a wrong approach can help eliminate pathways to a solution, but might also help me see possible tools to use in a solution. And in fact, this idea of using that “cut line” and similar triangles was important in my pathway to the end.
So when I went back to the drawing board, I wanted to really see how this diagram worked… Some things were fixed (the 12 by 19 rectangle, the fact that the semicircles sort of “slid,” and importantly, the fact that the semi-circles were tangent to the rectangle at two places). So I decided to build this diagram in geogebra (with only one of the semi-circles), and as I built it, I saw that everything hinged on the movable point “G.”
I made the line where the semi-circles touched movable, based on the location of point G. Play around with moving point G here on this web-based geogebra page, and try to get it so the semi-circle on the bottom is tangent to the right and bottom side of the rectangle!
So to me, everything hinged on location of point G, or in other words, the distance from A to G (which is the same as the distance from H to C). We are looking for the location of point G which makes the semi-circle perfectly tangent to right and bottom sides of the rectangle. So to me, those appeared to me as “keys” to the problem. 
Sooooo I drew my diagram, and importantly labeled the distance from point A to point G with a variable, a. And then I labeled lots of things in my diagram in terms of that variable and the radius of the semi-circle, r.
I had two variables, so I needed two independent equations. And here is something nice… because I initially went down a wrong path earlier with my mis-drawing, I had already gotten similar triangles in my head! So I got one equation from that.
I hunted and hunted, and found another equation I could get… using the Pythagorean Theorem!
So now I had two equations and two variables.
… and since I knew this was going to be a beast to solve, I just used Desmos, and got that the solution is a=1.5 and r=7.5.
I did a little of the algebraic gymnastics to try to work this out by hand, but it was pretty uninteresting to me and I was pretty convinced that if I really wanted to, I could. To me, getting the equations was the interesting part, and the rest felt like pencil-pushing. So I stopped there. It was nice that the geogebra applet I created seemed to confirm my answer for me:
So that was my process to solving this mathematical puzzle. Who knows – I could also be totally wrong! I’m left thinking of the following:
(1) Is there a more elegant way to come up with the answer? Because the answer is so nice (a diameter of 15?!?!) but it comes out of such an ugly set of equations, I bet there is a nicer way. In other words, is there a better “conceptual” approach that gives a stronger insight into the geometric nature of the setup?
(2) How did Matt come up with this puzzle? How did he come up with the 12 and 19, so that the answer worked out so neatly to a diameter of 15 (radius of 7.5)? Based on my playing around with this puzzle, I wouldn’t have expected a nice answer — so that shocked me. I would have anticipated nice side lengths and an ugly diameter, or ugly side lengths and a nice diameter.
Finally: If you like puzzles like this, you might want to google “Sangaku” and look at the twitter feed of Catriona Agg.
 At this point, I had a small detour where I briefly tried to work this problem on a coordinate plane, where I was finding the intersection of the two lines to find the location of the center of the circle, point I, based on the coordinates of G… but when I realized that once I had the intersection point, I’d still have to find find the right coordinates for G to make the circle tangent to the edges, I realized that would be annoying. So I abandoned the coordinate plane work, though I could always return to it if I needed.