Let c be the inscribed circle of the triangle ABC, d a line tangent 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′AA′A1=B′BB′B1=C′CC′C1 geometry unsolvedgeometry