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).

Answer 1

\[\angle ADB = 90° \left( \text{ Angle in a semicircle } \right)\]
\[\angle ADC = 90° \left( \text{ Angle in a semicircle } \right)\]
\[\text{ So } , \angle ADB + \angle ADC = 90° + 90° = 180\]
\[\text{ Therefore, BDC is a line }  . \]
\[\text{ Hence, the point of intersection of two circles lie on the third side }  .\]