Question

Two circles cut each other at A and B. A straight line CAD meets the circles at C and D. If the tangents at C and D intersect at E, prove that C, E, D, B lie on a circle.

Answer 1

Join A and B, B and C, B and D. 

In ∆ CDE, 

∠1 + ∠2 + ∠CED = 180°  ...(1)

∵ CE is a tangent to the circle CBA at point C, 

∠CBA is an angle in the alternate segment.

∴ ∠1 = ∠3 ...(2) 

∠2 = ∠4 ...(3) 

From (1), (2) and (3) we have 

∠3 + ∠4 + ∠CED = 180°

⇒ ∠CBD + ∠CED = 180° 

⇒ C, B, D, E are concyclic