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.


This isn‘t really an answer but gives information that might be of interest to you. The problem you quote is an example of a class which goes under the heading „recovering a triangle from three parts“. There are two types—-recovering from special points and recovering from quantities. The titles are self-explanatory and the one you mention belongs to the second type. There are three questions of interest—-the existence of a solution, its uniqueness, and its constructibility (in the sense of a ruler and compass construction). These questions have had a prominent position at varios epochs——in the nineteenth century during the resurgence of triangle geometry; and recently (applications of computers for finding and proving results). Their rebirth in modern times goes back to Euler, who considered some examples in a paper with the title „A simple proof of some difficult geometrical problems“ (my translation from the latin).

There are 27 possible choices of the quantities you mention for a given triangle and so the question has several variants (not 27, of course). There is a systematic method to reduce such questions to ones about equations in two or three variables—-again about existence, uniqueness and constructibility. I have done this explicitly for one of your cases and here it is, for what it is worth: the equations are $b^2-p^2=p^2 B$ and $$A(a+b)^2 =((b-ap-bp)^2+(a+b)^2(b^2-p^2). $$ Here, these are equations in the variables $a$ and $b$, where $A>0$ and $B>0$ are given and $p=\frac 12(b^2-a^2+1)$.

  • $\begingroup$ Thanks! This is useful. What are the geometric interpretations of $a$, $b$, $A$ and $B$? Presumably if we try to solve for $a$ and $b$, we need to do more than take square roots (this might explain Yau's comment about non-constructibility)? $\endgroup$ – Timothy Chow May 16 '19 at 15:02
  • $\begingroup$ $a$ and $b$ are side lengths. $A$ and $B$ are given by the data—-the squares of the tangent of the given angle and the lenght of the trisector. I assume that the given side length is 1. I tried to solve this with mathematica. The solution was rather huge and toocomplicated to see at a glance if it is constructible. $\endgroup$ – user131781 May 17 '19 at 4:17

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