An equilateral triangle inside a rectangle. What is red : yellow : blue?

# Category: Intermediate

## Mountain roads

A circular mountain peak has coordinates (0,0,3) and its base has towns C=(-4,0,0), E=( 4,0,0) on opposite sides. Town D=(0,4,0) is halfway between them, around the base. A road goes partially up and down the mountain connecting C and E. Another road connects C and D. Both roads are shortest possible. What is the ratio of the road lengths?

## The same vein

Two similar triangles ABC and EBD. Two line segments meet at point F. Prove that ABFC and EBDF are cyclic quadrilaterals.

## The pentagon proportion

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

## In touch

A regular nonagon and four equilateral triangles. The purple and yellow triangles share a vertex. Prove that the yellow triangles also share a vertex.

## The party hat II

An equilateral triangle with a circular arc through its red centre and two vertices. If the yellow area equals the green one, what’s the angle α?

## Holes in my heart

The heart is divided into right triangles, each of which contains a blue circle. What is the total circumference of the blue circles in relation to the green perimeter and the lengths of the purple and orange lines in the centre?

## Three times three

A regular nonagon and two equilateral triangles. What’s the angle α?

## The hex division

A regular hexagon with two diagonals and two perpendiculars. What’s red : yellow?

## The owl of Ceva

A triangle with three concurring cevians. What’s orange : blue?