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Beginner

The remote corner

A square and a circle. What is the green area?

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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.

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Intermediate

Seeing double

Five circles, of which the green circles are congruent and so are the blue ones. If rblue : rgreen= 6, what’s rred : rgreen?

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Intermediate

Égalité

A square with its diagonal and a circle. If red is equal to blue, what’s the angle α?

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Intermediate

House of Orange

A right triangle with a red and yellow square attached. Two circular arcs centered in the triangle vertices. What is the area of the orange rectangle in terms of red and yellow?

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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?

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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.

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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
Intermediate

As within, so without

A semicircle and a circle are tangent. The tangency point is shown. Prove that the two coloured triangles are similar.

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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?