Categories
Beginner

The leaning diamond

A square with two equilateral triangles. What’s the angle α?

Categories
Advanced

Cramped space

A triangle with three circles, two of which are tangent. Prove that the upper vertex and the two tangent points are collinear.

Categories
Intermediate

The dunce cap

The orthocentre H, circumcentre O, the incentre I, and points B and C all lie on the top of the dunce’s spherical head. What is the angle α?

Categories
Intermediate

The red dot

A right triangle with its incircle. What is the red fraction?

Categories
Intermediate

Locus pocus

The blue vertex is fixed in the plane. The green vertex can be anywhere on a circle. The yellow vertex completes an equilateral triangle. What is the locus of the yellow vertex?

Categories
Intermediate

Folding to pyramid

A square and an equilateral triangle. The square is folded as shown. What is the angle α?

Categories
Intermediate

Where is the other circle?

Given a point A on a circle and another point I in the interior of the circle, find two other points B, C on the circle so that I is the incentre of the triangle ABC. Find B and C as the intersection points between the given circle and some other circle.

Categories
Beginner

Outside the pentagon II

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

Categories
Beginner

The boot

Two squares and some line segments. Find the relation between the angles α and β.

Categories
Intermediate

The belly button

A quarter circle with a right triangle. Prove that the red point is the incentre of this triangle.