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).

Solution

Given: In ΔABC, 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 ⊥ BC.

Proof : ∵AD⊥BC

∴∠ADB=∠ADC=90o

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.

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