Categories
Advanced

The sneak peek

A rectangle with given side lengths is folded back and forth such that the corner touches the opposite side as shown. What is the minimal shaded fraction?

Categories
Advanced

The golf ball

A circle and a triangle with two sides tangent to the circle and a side connecting the tangency points. The line segment of the circle centre to the triangle apex is shown. Prove that α = β.

Categories
Advanced

The rotary movement

Two arbitrarily placed congruent equilateral triangles. Out of all the possible rotations of the plane, for how many will the image of ABC be A’B’C’?

Categories
Advanced

Spiegel im Spiegel

A triangle with an inscribed triangle. What fraction of the former does the latter cover in terms of lengths a, b, c, d, e and f?

Categories
Advanced

Out of proportion

An equilateral triangle containing a parallelogram, another equilateral triangle and a circular arc that is tangent in its base vertices. What is the proportion pink : blue?

Categories
Advanced

Circular architecture

An equilateral triangle and two circles. The centre of the right circle is the triangle side midpoint. Both circle intersections lie on a triangle side. Prove that the three red circular arcs are congruent.

Categories
Advanced

Out of focus

A semicircle with an inscribed triangle. Both circles are tangent to the semicircle, the diameter and one other triangle side. Prove that the line segment connecting the two shown tangency points is parallel to the diameter.

Categories
Advanced

The sacrosanct circles

Two tangent circles and two tangent line segments meeting in a point on the outer circle. The tangency points are connected by a line segment of length x. What’s x in terms of a and b?

Categories
Advanced

Extreme parallelism

Four parallel line segments, one of which is divided in three parts. What is the proportion blue : red : green?

Categories
Advanced

Silent circle

A large triangle with one side tangent to a circle. The tangency point is the vertex of a blue parallelogram. What is the red area?