Two equilateral triangles on a line. Prove that the blue triangle is equilateral as well.

# Category: Intermediate

## Tipping point II

Two regular pentagons share a vertex. What’s the angle α?

## All connected

Four coloured rectangles and three line segments. Prove that the three line segments are concurrent.

## Ten to nine

Two triangles and one incircle with its centre and tangency points. Prove that the triangles are similar.

## Quad stretches

Two squares sharing a vertex and two overlapping coloured quadrilaterals from square side midpoints. What is the area proportion of the blue and the red quadrilateral?

## Cathedral on the hills

A cathedral is erected on two hills, the side circular arcs whose centers are the hill ends, and so that the right (and left) side arcs are orthogonal. Show that the tip of the spire is directly above where the hills meet.

## Tangent latitudes

Tangent lines QC and EC meet at C. A point D on QC has DC=1 and QD=2. The line ED intersects the circle at G, and the line HGI is parallel to QDC. What is HG/GI?

## Square mouse

A semicircle and a square with extended side and diagonal. Prove that the red line segment is tangent to the semicircle.

## Touch the sides

What is the area of a green quadrilateral that fits inside a quarter circle, and has perpendicular diagonals?

## Ice cream cone II

Two regular pentagons sharing a vertex with extended sides and diagonals. Proof that the three red points are collinear.