The figure shows a right triangle ACB, BH is the altitude drawn to the hypotenuse, HE is perpendicular to BC, HF is perpendicular to AC, and EM and FN are perpendicular to AB. If BC = a, AC = b, AB = c, and CH = h, prove that
(1) , (2) , (3) , (4) , (5) HM = HN, and (6) , twice the geometric mean of AN and BM.