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Double eclipse

Given green and blue discs, construct a red region so that for every ray leaving L stays in the blue region exactly as long as it does in the pink region. (Namely, KM = OL.)

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Intermediate

The belly button

A quarter circle with a right triangle. Prove that the red point is the incentre of this triangle.

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Beginner

Steam roller

Two intersecting circles. Express the lengths b and the common tangent d in terms of c (the distance between the centres) and the angles β and γ.

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Intermediate

Under the dome

Two semicircles with four squares and a yellow rectangle. What is the aspect ratio of the latter?

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Intermediate

Pull the string

Two regular hexagons and a circumcircle. What is blue : red?

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Beginner

Fair scoop

What is the ratio of areas purple : orange if the cone angle is 45°, and the circular arc is a semicircle?

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Advanced

Mother’s umbrella

A circle containing two internally tangent circles. Prove that the red triangle is isosceles.

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Intermediate

Antennas above and below

The triangle ABC has orthocentre H, so that HA is an antenna above the green hill. The rectangle BCDE is inscribed in the circumcircle (ABC). Show that HA = CD = BE.

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Advanced

Crashing the kite

The orange lines are parallel, the purple lines are parallel, the dotted gray lines intersect the base at equal distances. What is the ratio of areas of the blue and green triangles?

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Intermediate

The doubleganger

Two nested right triangles with their incircles. Two tangency points are shown. Prove that the red and yellow line segments are parallel.