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The hippodrome shadow

The racer at A is running counter-clockwise around the inside of the outer purple circle, with a “shadow racer” B following behind, related by the three chords that are tangent to the inner orange circle. Does the shadow B ever catch up with A?

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Circle triplet

Identical circles have their centers aligned as shown, with the middle circle inscribed in a triangle. Show that the outer circles can be made tangent to the green circumcircle if the red axis is rotated correctly.

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Colour convergence

A green line connects two points of tangency of a triangle to its incircle. A red line is the angle bisector at a vertex, and an orange line connects the midpoints of two sides. Show that the three lines are concurrent.

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Alien eye

Two quarter-circles fit inside a square, and five circles fit inside the overlapped region, centred. What is the ratio of the (straight line) segment lengths red to blue? (The segments connect to points of tangency.)

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Shots in the dome

A beam from the top of one tower cuts through the dome, reflects off the ground, and hits the top of the other tower. Show that the line connecting the orange dots, the line connecting the blue dots, and the ground line are concurrent.

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The ceiling lamp

A regular octagon with two diagonals and a square. Prove that the four red points are concyclic.

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Seeing double II

A right triangle is divided in two triangles by an altitude. The three incircles are shown with three tangency points. Prove that the two red line segments are congruent.

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Peeping out II

A unit square and a rectangle touching three square sides and passing through the midpoint of the upper side. What is the minimal blue area?

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Blue convergence

Points B, C, D are on a circle with centre O and diameter COC’. Point E is on the line BC such that DE is perpendicular to COC’. Show that the perpendicular bisectors of EB and ED and the line DC’ are concurrent.

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The open roof

A regular hexagon with two line segments at right angles. What is the minimum of yellow : red?