Height AA1 CC1 acute-angled triangle ABC intersect at point E. prove that the angles cc1 a1 caa1 are equal.

Since AA1 and CC1 are heights, triangles AEC1 and CEA1 are rectangular

Right-angled triangles AEC1 and CEA1 are similar in acute angle since the angle AEC1 = CEA1 as vertical angles.

Then from the similarity of triangles: EC1 / EA1 = AE / CE.

In triangles EA1C1 and EAC, the angle A1EC1 = AEC as vertical angles, and EC1 / EA1 = AE / CE.

Then the triangles EA1C1 and EFS are similar in two proportional sides and the angle between them, which means the angle CC1A = CAA1, which was required to be proved.