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## Pentagon pleasure

Two regular pentagons. The centre of the large one is a vertex of the small one. What is yellow : green?

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## 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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## Pentagon offspring

Three regular pentagons are placed as shown. Prove that the three black line segments are concurrent.

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## Playing fields

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

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## Folding a pentagon

One side of a regular pentagon is folded back and forth as shown. Are the six red line segments congruent?

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## The telescope

The odd regular polygons are stacked together, in increasing order, zigzagging to form a telescope that extends forever! What is the limiting angle (with respect to the horizontal) at which the telescope points?

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## Outside the pentagon

A regular pentagon and two coloured equilateral triangles. What is the angle α?

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## The pentagon proportion

A regular pentagon and a circular arc through its red centre. What’s the proportion α : β?

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## The crosshairs

A regular pentagon and two semicircles. Prove that they are orthogonal, i.e. the tangents in the intersection point are orthogonal.

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## The vanilla ice cream

A regular pentagon with two extended sides and a right angle. What is blue : green?