Two quarter-circles fit inside a square, and five circles fit inside the overlapped region, centred. What is the ratio of the (straight line) segment lengths red to blue? (The segments connect to points of tangency.)

# Category: Advanced

## Shots in the dome

A beam from the top of one tower cuts through the dome, reflects off the ground, and hits the top of the other tower. Show that the line connecting the orange dots, the line connecting the blue dots, and the ground line are concurrent.

## The ceiling lamp

A regular octagon with two diagonals and a square. Prove that the four red points are concyclic.

## Seeing double II

A right triangle is divided in two triangles by an altitude. The three incircles are shown with three tangency points. Prove that the two red line segments are congruent.

## Peeping out II

A unit square and a rectangle touching three square sides and passing through the midpoint of the upper side. What is the minimal blue area?

## Blue convergence

Points B, C, D are on a circle with centre O and diameter COC’. Point E is on the line BC such that DE is perpendicular to COC’. Show that the perpendicular bisectors of EB and ED and the line DC’ are concurrent.

## The open roof

A regular hexagon with two line segments at right angles. What is the minimum of yellow : red?

## Highway confusion

Show that the green dotted line, which is an altitude of the blue triangle, is also a median of the orange triangle.

## The mis-stacked trays

A rectangle is turned 90°, bisecting a bigger rectangle of the same shape. When squares are removed from the two side pieces, the remaining rectangles have the same shape as well. What is the common ratio of the rectangle side lengths?

## Through the roof

A semicircle with its centre. What is green : red in terms of the angles α and β?