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Functional Geometry

Source: 2019 ISL G8

September 22, 2020
geometryIMO ShortlistIMO Shortlist 2019functional-geometry

Problem Statement

Let L\mathcal L be the set of all lines in the plane and let ff be a function that assigns to each line ℓ∈L\ell\in\mathcal L a point f(ℓ)f(\ell) on ℓ\ell. Suppose that for any point XX, and for any three lines ℓ1,ℓ2,ℓ3\ell_1,\ell_2,\ell_3 passing through XX, the points f(ℓ1),f(ℓ2),f(ℓ3)f(\ell_1),f(\ell_2),f(\ell_3), and XX lie on a circle. Prove that there is a unique point PP such that f(ℓ)=Pf(\ell)=P for any line ℓ\ell passing through PP.
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