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Advanced

Pentagon pleasure

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

Categories
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.)

Categories
Intermediate

Pentagon offspring

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

Categories
Intermediate

Playing fields

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

Categories
Beginner

Folding a pentagon

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

Categories
Intermediate

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?

Categories
Beginner

Outside the pentagon

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

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Intermediate

The pentagon proportion

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

Categories
Advanced

The crosshairs

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

Categories
Advanced

The vanilla ice cream

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