geometry puzzles

Category: Intermediate

A square deal

A regular hexagon and a square share a vertex. What’s the angle α?

A circle with a red and a green tangent. The two tangency points are shown. Prove that the blue and red line segments are parallel.

Two congruent rectangles inside a square. What fraction of the square is the rectangle overlap?

Two intersecting circles and a common tangent. The two tangency points are shown in black. Prove that the three red points are collinear.

A regular pentagon and a circle through two of its vertices and its centre. What’s the angle α?

Three tangent semicircles and two coloured triangles. What’s the proportion yellow : green?

A non-isosceles triangle with a red cevian and three line segments to a point on the cevian. Prove that this point is the incentre of the triangle.

A regular hexagon shares a vertex with an equilateral triangle of area 7. What is the area of the hexagon?

A regular hexagon with a diagonal and a circular arc centred in a side midpoint. What fraction is yellow?

A regular pentagon with an extended side. What’s the angle α?