Question

Two circle with centres O and O' are drawn to intersect each other at points A and B. Centre O of one circle lies on the circumference of the other circle and CD is drawn tangent to the circle with centre O' at A. Prove that OA bisects angle BAC.

Answer 1

Join OA, OB, O'A, O'B and O'O.

CD is the tangent and AO is the chord.

∠OAC = ∠OBA  ...(Angles in alternate segment)

In ΔOAB,

OA = OB  ...(Radii of the same circle)

∴ OAB = ∠OBA  ...(ii)

From (i) and (ii)

∠OAC = ∠OAB

Therefore, OA is bisector of ∠BAC