IMO 1984 LL AUS3

Origin: AUS

Problem

The opposite sides of the reentrant hexagon AFBDCE in- tersect at the points K, L, M (as shown in the figure). It is given that AL = AM = a, BM = BK = b, CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f. (a) Given length a and the three angles \alpha, \beta, and \gamma at the vertices A, B, and C, respectively, satisfying the condition \alpha + \beta + \gamma < 180◦, show that all the angles and sides of the hexagon are thereby uniquely determined. (b) Prove that a + 1 e = 1 b + 1 d. Easier version of (b). Prove that (a + f)(b + d)(c + e) = (a + e)(b + f)(c + d).       D D D D D DD

 e e e e e e P P P P P P P A B C D E F K L M