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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 sidesABandACof the triangleABCas diameters. The circles intersected atD.Construction:ADis joined.To prove:Dlies onBC. We have to prove thatBDCis a straight line.Proof:∠ADB=∠ADC=90°...Angle in the semi circleNow,∠ADB+∠ADC=180°⇒∠BDCis straight line.Thus,Dlies on theBC.