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

# Category: Intermediate

## Pandora’s box II

Five squares and a diagonal. What fraction of the largest square is coloured?

## What’s your angle?

Six squares are placed as shown. What’s the angle α?

## 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?

## Under the dome

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

## Pull the string

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

## 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.

## Six six four

Two regular hexagons and a square. What’s the angle α?

## 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.)

## Pentagon offspring

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