The party hat II

An equilateral triangle with a circular arc through its red centre and two vertices. If the yellow area equals the green one, what’s the angle α?

Scroll down for a solution to this problem.


The angle α is 60°.

Extend the circle and draw the inscribed regular hexagon. It is easy to see that ABCD is a parallelogram. From the condition yellow is green it follows that the extended common base DE cuts the diagonal AC in half. So it must be the other diagonal DB. Now, ∠BEC=∠BAC=120°, making α=60°.

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