Question

Circles are described on the sides of a triangle as diameters. Prove that the circles on any two sides intersect each other on the third side (or third side produced).

Answer 1

Given: In $\Delta ABC$, circles are drawn with sides AB and AC as diameter.

To prove: Circles drawn on AB and AC intersect at D which lies on BC, the third side.

Construction: Draw AD $\perp$ BC.

Proof : $\because AD\perp BC$

$\therefore \angle ADB=\angle ADC={90}^o$

So, the circles drawn on sides AB and AC as diameter will pass through D.

Hence, circles drawn on two sides of a triangle pass through D, which lies on the third side.