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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?

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Beginner

Colour confinement

A semicircle with three tangent line segments. What is its area in terms of lengths a and b?

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Beginner

The Sikh turban

A regular hexagon containing an orange quadrilateral. What fraction of the total area is blue?

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Advanced

The green quad

Three semicircles and a circle all touching each other. The diameters of the two smallest semicircles are given. The large triangle connects three tangency points. What is the area of the green quadrilateral?

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Intermediate

The colour purple

A rectangle is split by two perpendicular lines. The four circles pass through vertices and intersection points. Their centres form a purple quadrilateral. What fraction of the large rectangle area does it cover?

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Beginner

The stained glass

Two congruent rectangles are placed side by side. Two line segments form two coloured quadrilaterals and three triangles. What is the proportion purple : green?