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## The square sail

A quadrilateral and a parallelogram share two sides, and a vertex from each determine the orange square. The lengths BC, EF, AD are 4,5,7 respectively, where E and F are midpoints of AB and DC. What is the orange area?

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## Eclipsing earth

Two circles with their centres. The quadrilateral ABCD is cyclic and has two sides tangent to the small circle. Prove that BD is also tangent.

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## Dancing squares

Four squares and one centre. Prove that the red quadrilateral is a parallelogram.

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## Triangles galore

Four equilateral triangles. Two centres are shown. Prove that the green quadrilateral is a parallelogram.

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## The same vein

Two similar triangles ABC and EBD. Two line segments meet at point F. Prove that ABFC and EBDF are cyclic quadrilaterals.

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## The boomerang

A quadrilateral and a triangle. What’s the angle α?

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## Baby steps

The edges of a cyclic quadrilateral extend to two intersections. O is the circle centre. The two diagonals intersect inside the circle. What is the angle α?

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## Pyramid force field

The blue quadrilateral contains a purple interior point which is a corner for two right angles. Reflect the purple point about the four sides of the quadrilateral. Show that the four image points are concyclic.

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## An odd wrapping

A present fits inside a circle, and we tape the orange ribbons to the green midpoints of the sides, choosing them perpendicular to the opposite sides at the purple points. Show that the four ribbons have a point in common.

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## Cycling of tangents

Four circles share one point. The quadrilateral has vertices which are intersections of pairs of circles, and three of its coloured edges are tangent to the circle of the same colour. Show that the fourth edge is also tangent to its circle.