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## Lining up half way

Show that the three red midpoints are collinear.

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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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## The crosshairs

A regular pentagon and two semicircles. Prove that they are orthogonal, i.e. the tangents in the intersection point are orthogonal.

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## The laser game

In an equilateral triangular mirror room a laser is shot from the top vertex down to the base. The light ray bounces three times to return to the same base point. What is the angle α?

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An orange circle is squeezed inside a rectangle. Show that no matter how the blue point moves on the circle, the derived points Q and P will satisfy |QL| = |KP|.

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## The vanilla ice cream

A regular pentagon with two extended sides and a right angle. What is blue : green?

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## Falling pillar II

Three congruent rectangles on a line. Prove that the four red points are cyclic.

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## The paper airplane

Express the area of the orange triangle in terms of the areas of the other colours.

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