Problem

Source: Stars of Mathematics 2007, Day 1, Problem 3

Tags: Pure geometry, geometry



Let $ ABC $ be a triangle and $ A_1,B_1,C_1 $ the projections of $ A,B,C $ on their opposite sides. Let $ A_2,A_3 $ be the projection of $ A_1 $ on $ AB, $ respectively on $ AC. B_2,B_3,C_2,C_3 $ are defined analogously. Moreover, $ A_4 $ is the intersection of $ B_2B_3 $ with $ C_2C_3; B_4, $ the intersection of $C_2C_3 $ with $ A_2A_3; C_4, $ the intersection of $ A_2A_3 $ with $ B_2B_3. $ Show that $ AA_4,BB_4 $ and $ CC_4 $ are concurrent.