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?

Scroll down for a solution to this problem.


No, B never catches up.

Solution by Matthew Arcus.

If B catches up, there is a triangle with inradius r, circumradius R, distance between centres d, so by Euler d² = R(R-r). But by continuity there is a triangle on XY with inradius r, so the incentre is also at distance d, but then the incircle is the same as the orange circle.

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