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An isosceles triangle with two inscribed congruent semicircles. What is the angle α?

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Solution

The angle α is 30°.

Solution by Rony Sarker.

First draw the radii CK and EJ to the tangency points. Because of the two right angles and equal radii, EC must be parallel to JK and hence ∠CEA=α.

Now draw a radius CA to the third tangency point and notice that right triangle ACE has sides of r and 2r. We find that sin(α)=1/2 and hence α is 30°.

Notice that we didn’t use the fact that the triangle is isosceles, so strictly speaking the problem is over-constrained: the top angle would still be 30° in a non-isosceles triangle.

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