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).

Given:

Circles are described on the sides of a triangle as diameters.

To do:

We have to prove that the circles on any two sides intersect each other on the third side (or third side produced).

Solution:

Let in a $\triangle ABC$, circles are drawn on sides $AB$ and $AC$.

Draw $AD \perp BC$

$AD \perp BC$

This implies,

$\angle ADB = \angle ADC = 90^o$

From the figure,

The circles drawn on sides $AB$ and $AC$ as diameters will pass through $D$.

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

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Updated on: 10-Oct-2022

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