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Beginner

The miniature version

Two rectangles share a vertex. Two line segments connecting opposing vertices and the common side intersect in a single point. Prove that the rectangles are similar.

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Intermediate

Wheels in motion

Two tangent circles and a third circle touching the line segment connecting their centres in the tangency point. Prove that the red points (centre and two intersection points) are collinear.

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Beginner

The Transamerica Pyramid

Two squares inscribed in a triangle. What fraction of its area is coloured?

Categories
Beginner

The three day stubble

A regular pentagon and a square share two vertices. What’s the angle α?

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Advanced

It’s complicated

A circle, a semicircle and several triangles. The black points are tangency points. What proportions have the line segments a : r : b?

Categories
Advanced

The odd couple

A semicircle with two inscribed triangles The tangency points of the incircle of the bigger triangle are shown. What is the triangle area proportion orange : blue?

Categories
Intermediate

Parallel universes

Two straight lines are specified by y=a+cx and y=b+cx, respectively, where b≥a. What’s the distance d between them in terms of a, b and c?

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Advanced

Blazing hot

A circle inscribed in a triangle inscribed in a semicircle. The line segment through two tangency points meets a radius at the semicircle circumference. What is the angle α?

Categories
Beginner

Trinity

Three squares are placed as shown. What fraction of the total area is green?

Categories
Intermediate

The turning point

Two isosceles triangles share a vertex. What is the maximum of their area proportion red/green?