A triangle has one vertex at a circle's center and two vertices on the circle. Can the three enclosed regions have rational areas?

geometry real-analysis trigonometry number-theory

Solution 1:

I think I can answer my own question. The sine of any rational multiple of $\pi$ is algebraic, as shown here, so it cannot be a rational multiple of $\pi$, so the answer to my question is no.

(I thought about this question on and off for a few days before posting it here. Then, for some reason, almost immediately after posting the question here, without receiving any replies, I found the answer.)

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