Problem (2015IranGOAp3): let $ H $ the orthocenter of the triangle $ ABC $ pass two lines $ l_1 $ and $ l_2 $ through $ H $ such that $ l_1 \bot l_2 $ we have $ l_1 \cap BC = D $ and $ l_1 \cap AB = Z $ also $ l_2 \cap BC = E $ and $ l_2 \cap AC = X $ like this picture pass a line $ d_1$ through $ D $ parallel to $ AC $ and another line $ d_2 $ through $ E $ parallel to $ AB $ let $ d_1 \cap d_2 = Y $ prove $ X $ $ , $ $ Y $ and $ Z $ are on a same line
AlphaGeometry formalization: a b c = triangle; h = orthocenter a b c; p = free; d = on_line b c, on_line h p; z = on_line a b, on_line h p; q = on_tline h h p; e = on_line b c, on_line h q; x = on_line a c, on_line h q; y = on_pline d a c, on_pline e a b ? coll x y z
HAGeo formalization: A B C = triangle; H = orthocenter A B C; l1 = line_through H; l2 = perpendicular_line H l1; BC = line B C; D = intersection l1 BC; AB = line A B; Z = intersection l1 AB; E = intersection l2 BC; AC = line A C; X = intersection l2 AC; m1 = parallel_line D AC; m2 = parallel_line E AB; Y = intersection m1 m2; Prove: collinear X Y Z
Difficulty score: 3.1