IMO 1967 Problem B1 A 0 B 0 C 0 and A 1 B 1 C 1 are acute-angled triangles. Construct the triangle ABC with the largest possible area which is circumscribed about A 0 B 0 C 0 (BC contains A 0 , CA contains B 0 , and AB contains C 0 ) and similar to A 1 B 1 C 1 .
Problem 992
Official solution
Take any triangle similar to A 1 B 1 C 1 and circumscribing A 0 B 0 C 0 . For example, take an arbitrary line through A 0 and then lines through B 0 and C 0 at the appropriate angles to the first line. Label the triangle's vertices X, Y, Z so that A 0 lies on YZ, B 0 on ZX, and C 0 on XY. Now any circumscribed ABC (labeled with the same convention) must have C on the circle through A 0 , B 0 and Z, because it has ∠C = ∠Z = ∠C 1 . Similarly it must have B on the circle through C 0 , A 0 and Y, and it must have A on the circle through B 0 , C 0 and X. Consider the side AB. It passes through C 0 . Its length is twice the projection of the line joining the centers of the two circles onto AB (because each center projects onto the midpoint of the part of AB that is a chord of its circle). But this projection is maximum when it is parallel to the line joining the two centers. The area is maximised when AB is maximised (because all the triangles are similar), so we take AB parallel to the line joining the centers. [Note, in passing, that this proves that the other sides must also be parallel to the lines joining the respective centers and hence that the three centers form a triangle similar to A 1 B 1 C 1 .] s are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48- X. 9th IMO 1967 © John Scholes [email protected] 29 Sep 1998 Last corrected/updated 27 Oct 2002