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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.
Solution
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
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