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Question

In the given figure two circles intersect each other in points A and B.Two tangents touch these circles in points P,Q and R,S as shown.line AB intersects seg PQ in C and seg RS in D.Show that C and D are the mid points of seg PQ and seg RS respectively.

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Solution

CP is a tangent and CD is a secant.By tangent secant theorem, we have:CP2=CA×CD ...(1)CQ is a tangent and CD is a secant.By tangent secant theorem, we have:CQ2=CA×CD ...(2)Using eqs. (1) and (2), we have:CP2=CQ2CP=CQ Hence, C is the midpoint of PQ.

RD is a tangent and DC is a secant.By tangent secant theorem, we have:RD2=DB×DC ...(3)CQ is a tangent and CD is a secant.By tangent secant theorem, we have:DS2=BD×DC ...(4)Using eqs. (3) and (4), we have:RD2=DS2RD=DS Hence, D is the midpoint of RS.

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