IMO 1977 LL GDR18

Origin: GDR

Problem

Given an isosceles triangle ABC with a right angle at C, construct the center M and radius r of a circle cutting on segments AB, BC, CA the segments DE, FG, and HK, respectively, such that \angleDME + \angleFMG + \angleHMK = 180◦and DE : FG : HK = AB : BC : CA.

Solution

Let U be the midpoint of the segment AB. The point M belongs to CU and CM = ( \sqrt 5 −1)CU/2, r = CU \sqrt 5 −2.