Mathematics
Two circles with centers O and O' are drawn to intersect each other at points A and B. Center O of one circle lies on the circumference of the other circle and CD is drawn tangent to the circle with center O' at A. Prove that OA bisects angle BAC.
Circles
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Answer
Join O'A and O'B.
As, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment, we have :
CD is the tangent and AO is the chord.
∴ ∠OAC = ∠OBA ……… (1)
In ∆OAB,
OA = OB [Radius of the circle with center O.]
∠OAB = ∠OBA ………. (2) [Angles opposite to equal sides]
From (1) and (2), we have
∠OAC = ∠OAB
Hence, proved that OA is the bisector of ∠BAC.
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