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The square of Amaresh

A right triangle having semicircles along its perpendicular sides and two smaller inscribed circles. Prove that the orange quadrilateral is a square.

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Beginner

The French kiss

Two tangent circles and two red tangent line segments. The three tangency points are shown. What is the angle β in terms of the angle α?

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Intermediate

Similar intention

A cyclic quadrilateral and two diagonals. What fraction of its area is blue?

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Intermediate

Z

A green quadrilateral of which two sides are tangent to a quarter circle of radius r. What is its area?

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Beginner

The flying saucer

A quarter circle with four chords. What is the proportion purple : gold?

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Advanced

The freight elevator

Four points A, B, C and D form a quadrilateral. Prove that it is a rectangle if and only if for an arbitrary point P we have PA2+PC2=PB2+PD2.

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Intermediate

Brothers in arms

The tangency points of two tangents of a circle are connected. Three perpendiculars are drawn from a point on the circle, of which two have a given length. What’s the proportion red : yellow?

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Beginner

The tornado

A quadrilateral covers exactly half of the area of a rectangle. Find the length x in terms of a, b and c.

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Intermediate

The tipping point

Two touching circles, two diameters and two common tangents. What is the area of the green quadrilateral in terms of radii x and y?

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Beginner

Square eyes

A square with four quarter circles of which the intersection points are the vertices of a yellow quadrilateral. What fraction of the total area is yellow?