I spent about 15 minutes noodling around. I tried adding different lines. I tried writing ratios, because there are so many similar triangles here. I had trouble using the digital annotation tools I was using, which made it harder to mark congruent angles. Eventually I started thinking about how I would get EF + DG as an expression at all, so I started thinking about making rectangles. Maybe their perimeter would somehow be useful?
I was feeling ready for a hint, so I turned to the back of the book for this hint:
“Draw CD, CE, and the altitude from C to AB; then prove triangles congruent.”
I quickly saw how this would be useful. I had already tried drawing these lines, but I hadn’t seen how it would be useful. Now I saw that if I could prove these triangles congruent, I would immediately have DE = EF + DG. I don’t know how I missed it earlier:
I didn’t have the angles yet. I was confident I would need to use the isosceles triangles that BD = BC and AE = AC created. Still, I was having trouble, and was feeling impatient. I decided to look at the solution.
That’s pretty much what I was expecting, though it’s a bit annoying that I didn’t get it easily. Was I just lazy? I was definitely a bit lazy. But I think there’s something else here worth remembering, which is that when the angles are all bunched up like this it can be useful to write the actual equations using the angles out. I think it would have been easier to see the relationships that way, as I had a hard time seeing all the relationships in the image. (I struggled to annotate the diagram in a way that contained this info after reading the solution.)
A very brief note: I tried to use a tablet to draw on the image, which sped up the process in some ways. It also makes me more reluctant to write out equations in a way that would have been helpful here. I used Google Jamboard for this problem, but I’d like to find something better.
Takeaways: don’t forget to write equations for angles; maybe do that on paper or find a better way to do geometry on the tablet.