Question

Prove that the circle drawn on any one of the equal sides of an isosceles triangle as diameter, bisects the third side of the triangle.

Answer 1

Let ABC is the isosceles triangle where AB = AC 
The circle drawn on AC as diameter intersects BC at D
From Circle theorem we know that the angle inscribed on semicircle is 90 degree
Hence angle ADC is 90 degree 
So we have angle ADB is 90 degree
This gives Δ ADC and Δ ADB are right angled triangle.
We have hypotenuse =AB=AC and AD common leg
Two right triangles are congruent if the hypotenuse and one corresponding leg are equal in both triangles.
as.. Δ ADC ΔADB are congruent
∴ BD=DC