Categories
Intermediate

The triple airbag

A triangle containing three tangent semicircles with diameters a, b and c. Three line segments pass through the red tangency points. What is the triangle perimeter?

Categories
Advanced

The grazing shot

A parabola and its directrix in red. Two tangents intersect at a point on the directrix. What’s the angle α?

Categories
Intermediate

The iron curtain

A parallelogram ABCD with a circle and a diameter CE. Prove that point E is the orthocentre of triangle ABD.

Categories
Intermediate

Worlds colliding

Two intersecting circles and a circle passing through both their centres and their intersection points. What’s the angle α?

Categories
Beginner

The common triangle

Two semicircles contain a triangle. The centre of the right semicircle is shown. Prove that the triangle is isosceles.

Categories
Intermediate

At navel level

Two equilateral triangles share a vertex. The centre of the larger one is aligned with the midpoint of a side of the smaller one. What is green : red?

Categories
Intermediate

Sharing is caring

Three equilateral triangles and three shared vertices. What’s the proportion blue : green?

Categories
Beginner

The fried egg

Two concentric circles and two tangents. What is the yellow fraction?

Categories
Intermediate

Lopsided

The apex of a parabola is the shared vertex of two rectangles. Proof that the rectangles are similar.

Categories
Intermediate

Harmonic proportions

A trapezium with its diagonals and a parallel line segment. Prove that yellow : purple = blue : red.