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Source: XVII Sharygin Correspondence Round P20

March 2, 2021
EulergeometrySharygin Geometry Olympiadfunctional equation

Problem Statement

The mapping ff assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.)
(a) Let σ\sigma be any similarity in the plane and let σ\sigma map triangle Δ1\Delta_1 onto triangle Δ2\Delta_2. Then σ\sigma also maps circle f(Δ1)f(\Delta_1) onto circle f(Δ2)f(\Delta_2).
(b) Let A,B,CA,B,C and DD be any four points in general position. Then circles f(ABC),f(BCD),f(CDA)f(ABC),f(BCD),f(CDA) and f(DAB)f(DAB) have a common point.
Prove that for any triangle Δ\Delta, the circle f(Δ)f(\Delta) is the Euler circle of Δ\Delta.
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