Wednesday , July 16 , 2008
Let H be the orthocenter of an acute-angled triangle ABC. The circle ΓA centered at the midpoint of BC and passing through H intersects the sideline BC at points A1 and A2. Similarly, define the points B1, B2, C1 and C2.
Prove that six points A1 , A2, B1, B2, C1 and C2 are concyclic.
(ii) Prove that equality case is achieved for infinitely many triples of rational numbers x, y and z.