At first, I just drew the diagram and stared at it for a while. I tried to mark congruent sides and angles. I didn’t really get anywhere.

Then, I thought that it would be a good idea to play with parallels. This was largely because of two things:

I knew those midpoints were in there, and midpoints sometimes create parallel lines

Because of the previous problem I worked on, I was thinking about how lines that are perpendicular to a line are also perpendicular to all of its parallels. I thought that would be good to play with again.

You can see my first sketch, where I was mostly staring at the diagram and playing with angles. Then I added another line, thinking that it would probably create congruent triangles.

It did!

This happens to be the solution the book names as well:

This really is connected to the previous problem I worked on. It pulled off the exact same “add a parallel and get yourself a free right angle” move. It was immensely satisfying to be able to see the connection between these two very different problems.

These connections are so important for learning, and so difficult to notice on your own. This is one reason why it is important to look back on solutions and compare them when self-studying — while working on the problem for the first time, even if you get the solution, you won’t necessarily see what links the solutions of different problems together.

I like using my blog as a journal, but it seems like it would be so much fun to find a way to dynamically link these solutions as I see them. A wiki? A static website that I edit over time? I don’t know, but it’s still fun to try problems and see the connections.

I’m also feeling motivated to review some of the solutions from problems I have already studied. Maybe I’ll make flashcards in Anki for them? It would be cool to be able to really think about the connections, and it’s hard to do that without a bunch of solutions in mind. If I were taking a class, at some point the teacher might ask us to study for an exam — that’s the sort of thing I’m thinking would be a useful addition to my learning mix.