Question

# If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Solution

## Given,Two circles are drawn on the sides $$AB$$ and $$AC$$ of the triangle $$ABC$$ as diameters. The circles intersected at $$D$$.Construction: $$AD$$ is joined.To prove: $$D$$ lies on $$BC$$. We have to prove that $$BDC$$ is a straight line.Proof:$$∠ADB = ∠ADC = 90°$$        ...Angle in the semi circleNow,$$∠ADB + ∠ADC = 180°$$$$⇒ ∠BDC$$ is straight line.Thus, $$D$$ lies on the $$BC$$.MathematicsRS AgarwalStandard IX

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