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## The leaning diamond

A square with two equilateral triangles. What’s the angle α?

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## Cramped space

A triangle with three circles, two of which are tangent. Prove that the upper vertex and the two tangent points are collinear.

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## The dunce cap

The orthocentre H, circumcentre O, the incentre I, and points B and C all lie on the top of the dunce’s spherical head. What is the angle α?

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## The red dot

A right triangle with its incircle. What is the red fraction?

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## Locus pocus

The blue vertex is fixed in the plane. The green vertex can be anywhere on a circle. The yellow vertex completes an equilateral triangle. What is the locus of the yellow vertex?

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## Folding to pyramid

A square and an equilateral triangle. The square is folded as shown. What is the angle α?

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## Where is the other circle?

Given a point A on a circle and another point I in the interior of the circle, find two other points B, C on the circle so that I is the incentre of the triangle ABC. Find B and C as the intersection points between the given circle and some other circle.

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## Outside the pentagon II

A regular pentagon and two equilateral triangles. What’s the angle α?

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## The boot

Two squares and some line segments. Find the relation between the angles α and β.

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## The belly button

A quarter circle with a right triangle. Prove that the red point is the incentre of this triangle.