Let L be the set of all lines in the plane and let f be a function that assigns to each line ℓ∈L a point f(ℓ) on ℓ. Suppose that for any point X, and for any three lines ℓ1,ℓ2,ℓ3 passing through X, the points f(ℓ1),f(ℓ2),f(ℓ3), and X lie on a circle.
Prove that there is a unique point P such that f(ℓ)=P for any line ℓ passing through P.Australia geometryIMO ShortlistIMO Shortlist 2019functional-geometry