IMO 1984 LL FRA18

Origin: FRA

Problem

Let c be the inscribed circle of the triangle ABC, d a line tan- gent to c which does not pass through the vertices of triangle ABC. Prove the existence of points A1, B1, C1, respectively, on the lines BC, CA, AB satisfying the following two properties: (i) Lines AA1, BB1, and CC1 are parallel. (ii) Lines AA1, BB1, and CC1 meet d respectively at points A′, B′, and C′ such that A′A1 A′A = B′B1 B′B = C′C1 C′C .