A square with its diagonal and a circle. If red is equal to blue, what’s the angle α?

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## Solution

The angle α is 30°.

This solution uses trigonometry. First, the length are calculated as seen above, using the angle bisector theorem and Angles Subtended by Same Arc Theorem. We us t=tan(α) and c=cos(α).

Now blue can be calculated as sin(90-α)t^{2}/(2c(t+1))=t^{2}/(2(t+1)). Similarly red equals sin(45-α)/(2c√2)=(1-t)/4. Equating the two expressions gives a quadratic equation in t, which has one positive solution t=1/√3 and hence α=30°.

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