In Shing-Tung Yau's autobiography The Shape of a Life, he mentions a problem that he came up with as a teenager.
Suppose you know the length of one side of a triangle, one angle, and the length of one angle bisector. Can you construct the corresponding triangle, using just a compass and ruler? I worked on this problem for the better part of a year and made little headway. …
One day, I found a book that discussed [this problem]. I learned that it could not be solved, which came as quite a relief. The book cited a recent argument that proved you could not construct one, and only one, triangle that satisfied three of these conditions.
I was excited to see that "my problem" had stumped other people and was only recently shown to be insoluble. I further realized that this same problem was similar to one that dated back many centuries: Could you trisect an angle if you had only a ruler and compass? No, you could not. Nor could you solve another long-standing problem, "squaring a circle." … I was proud to find out that my problem was in the same category as these two classic problems.
I'm curious to learn more about the history and literature of this problem. I tried doing some searching but when I use the obvious keywords, I get too many hits on unrelated problems.
Possibly this question belongs on some other stackexchange site; I'm willing to migrate it if people think it should be.
EDIT: I tried contacting Yau directly. He confirmed that there is indeed a non-constructibility theorem here, but he was not able to come up with a literature reference on the spot. In particular he couldn't remember much about the book where he first saw the result in print, other than that it was in Japanese.