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Here is one elementary, yet interesting geometric question.

If we have a right triangle, its circumcenter is the midpoint of its hypotenuse.

How to prove other direction? Namely,

Is there a non-right triangle which corresponding circumcenter is lying on one of its sides?

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    No, by the Theorem of Thales – lulu Aug 21 '20 at 23:52
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    No, the side with the circumcenter would be a diameter of the circle. Then the third vertex of the triangle (the one that isn't an endpoint of the diameter) would necessarily be a right angle, to subtend an arc of 180 degrees. – DreiCleaner Aug 21 '20 at 23:54
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    Circumcenter location can be used to classify triangles. If it is on a side, the triangle has a right angle. If it is outside the triangle, the triangle has an obtuse angle. If it is inside the triangle, the triangle has all acute angles. – DreiCleaner Aug 21 '20 at 23:59
  • Thanks, but I am interested in rigorous and detailed geometric proof – 1b3b Aug 22 '20 at 00:07
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    This is literally the Theorem of Thales. The Wiki article I linked to contains three proofs. – lulu Aug 22 '20 at 00:08
  • Okay this indeed is Theorem of Tales, but a bit disguised. Thanks – 1b3b Aug 22 '20 at 00:18

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