Question

In the figure, AB is common chord of the two circles. If AC and AD are diameters; prove that D, B and C are in a straight line. O₁ and O₂ are the centers of two circles.

Answer 1

Given: Two circles with centre O₁ and O₂ intersect each other at A and B. AC and AD are the diameters of the circles.

To prove: D, B, C are in the same straight line.

Construction: Join AB.

Proof:

AO₁C is diameter.

∠ABC = 90°    ...(Angle in a semi-circle)

Similarly, ∠ABD = 90°,

Adding, we get:

∠ABC + ∠ABD = 90° + 90° = 180°

DBC is a straight line or D, B, C are in the same line.