Categories
Intermediate

The doubleganger

Two nested right triangles with their incircles. Two tangency points are shown. Prove that the red and yellow line segments are parallel.

Categories
Intermediate

Sticky corners

Two equilateral triangles and several line segments. What is red : yellow?

Categories
Intermediate

Median between sides

A triangle has sides a,b,c and the medians d,e,f ending at those sides, respectively. Given the lengths (a,b,f) construct the triangle. (Start with a triangle with side lengths determined in some manor from the given 3 lengths.)

Categories
Beginner

The punctured hexagon

A regular hexagon, a square and an equilateral triangle. Show that the top vertex of the triangle is on the hexagon circumcircle.

Categories
Advanced

Circle triplet

Identical circles have their centers aligned as shown, with the middle circle inscribed in a triangle. Show that the outer circles can be made tangent to the green circumcircle if the red axis is rotated correctly.

Categories
Intermediate

Wedge the wedge

An equilateral triangle containing an isosceles triangle that is not equilateral. What’s the angle α?

Categories
Intermediate

Vanishing point II

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

Categories
Beginner

Two-story A-frame house

The slanted lines are perpendicular, and the one in the bottom story bisects the angle. Show that red equals green, so the stories have the same height.

Categories
Advanced

Colour convergence

A green line connects two points of tangency of a triangle to its incircle. A red line is the angle bisector at a vertex, and an orange line connects the midpoints of two sides. Show that the three lines are concurrent.

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.