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Two moons

A rectangle with two congruent circles, a semicircle and a common tangent. What is the proportion of side lengths a : b?

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Round and pointy

A circle is inscribed in both a semicircle and a right triangle. The three tangency points are shown. What is the angle α?

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Elasticity

Four squares and two triangles. Prove that the two green angles are congruent.

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It’s complicated

A circle, a semicircle and several triangles. The black points are tangency points. What proportions have the line segments a : r : b?

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The odd couple

A semicircle with two inscribed triangles The tangency points of the incircle of the bigger triangle are shown. What is the triangle area proportion orange : blue?

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Blazing hot

A circle inscribed in a triangle inscribed in a semicircle. The line segment through two tangency points meets a radius at the semicircle circumference. What is the angle α?

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Camera obscura

A quarter circular room is covered with mirrors. A ray of light is emitted from one corner and reflected somewhere on the opposite wall, then on the arc, before reaching the adjacent wall at a point a distance a from the light source and b from the circle centre. What is the maximum of a/b?

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The freight elevator

Four points A, B, C and D form a quadrilateral. Prove that it is a rectangle if and only if for an arbitrary point P we have PA2+PC2=PB2+PD2.

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The Eightfold Path

A semicircle, a square and an isosceles triangle. What’s the angle α?

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Fun with flags

A triangle with two cevians. What is d/c-b/a?