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Intermediate

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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Video: Folding to pyramid

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Intermediate

The red dot

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

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Intermediate

Hidden parallelogram

Blue circle (ACDGB) with CEG || AB. Pink circle (EDG) intersects AB at I. Extend DI to intersect blue circle at J, then JC intersects AB at M. Show that IM = CE.

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Video: Outside the pentagon II

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Beginner

Four five

A square and a regular pentagon share a vertex. What’s the angle α?

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Video: Stars on 45

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Intermediate

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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Intermediate

Orthogonal partial eclipse

Two circles are orthogonal (tangents at intersections are perpendicular). How must the points A and B be related?

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Video: Staying in line

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