Categories
Intermediate

Blowing bubbles

Three congruent blue circles with a common intersection point. Prove that the red circle through the other intersections is also congruent.

Categories
Intermediate

Vanishing point II

Six squares and four coloured triangles. What is red : blue : yellow : green?

Categories
Intermediate

Balancing act II

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

Categories
Intermediate

Mind the steps

Three similar rectangles and a line segment. Express red in terms of blue and green.

Categories
Intermediate

In and out

An obtuse triangle, of which α is the obtuse angle, with its incircle, two cevians and two tangents. Find a relation between α and β.

Categories
Intermediate

Pop you bubble

A square with a semicircle and a circle of equal radius. Their tangency point is shown. Prove that the red triangle is equilateral.

Categories
Intermediate

Wheel of fortune II

A triangle with its incircle and three cevians. The tangency points and incentre are shown. Find the relation between the angles α, β and γ.

Categories
Intermediate

In times out

What is the area of the green rectangle, which has sides tangent to the incircle and circumcircle of a triangle with side lengths a, b, c.? (Write the answer as a multiple of a quotient of two elementary symmetric functions in a, b, c.)

Categories
Intermediate

Four six four

A regular hexagon and two similar rectangles. What fraction is green?

Categories
Intermediate

Eclipsing earth

Two circles with their centres. The quadrilateral ABCD is cyclic and has two sides tangent to the small circle. Prove that BD is also tangent.