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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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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

The square sail

A quadrilateral and a parallelogram share two sides, and a vertex from each determine the orange square. The lengths BC, EF, AD are 4,5,7 respectively, where E and F are midpoints of AB and DC. What is the orange area?

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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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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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Intermediate

Pentagon divided

Given a pentagon ABCDE, construct the points T,R,Q,U such that the segments AT, AR, AQ, and AU divide the pentagon into five equal area pieces. (Using straight edge and compass. But you may divide a segment into a number of equal segments for free.)

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Advanced

Two medians and one side

A triangle has sides a,b,c and the medians d,e,f ending at those sides, respectively. Given only the lengths (b,d,e) construct the triangle (a,b,c). (Start with a triangle with side lengths determined in some manor from the given 3 lengths.)

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Intermediate

Wings

Two wings are attached to the corner of a rectangle, and have corners lined up with the diagonal of the rectangle. The angles at the lower wing tips are equal. Must α = β at the left wing tips?

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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

Playing fields

Show equality of the green and orange square playing field areas.