Two circles touch internally at M. A straight line touches the inner circle at P and cuts the outer circle at Q and R. Prove that Angle QMP = Angle RMP..

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Construction:Draw a tangent to the circle at their point of contactSolution:Let SMQ = ythenSMQ = MRQ   Alternate segment theoremLet TPR = x then TPR = TMP= x     Alternate segment theorem.....(1)MPQ = x + y    Exterior angle of Δ RMPMPQ = MTP = x + y    Alternate segment theorem MTP = PMS = x+ y     Alternate segment theoremQMP = SMP -SMQ= x+ y y= x.......2From 1 and 2 we get TMP = QMP = xHence proved.....

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