Question

Prove that the *** centre of the circles described on the sides of a triangle as diameters is the orthocentre of the triangle.

Answer 1

The circles described on AB and AC as diameters (see Fig.) will intersect at a point D on the side BC and because ΔADB = ΔADC =- 90°.

We have that AD is an altitude of ΔABC which is also the ** axis of the two circles with AB and AC as diameters. Similarly, the other altitudes BE and CF are the ** axes of the pairs described on AB, BC and AC, BC. Thus, the altitudes of ΔABC are ** axes. Hence, the orthocentre is their ** centre.