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Math Help - Two circles intersecting each other...

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    Two circles intersecting each other...

    Two circles intersect each other at A and B as shown in the figure. The common chord AB is produced to meet a common tangent PQ to the circles at D. Prove that DP=DQ.
    Attached Thumbnails Attached Thumbnails Two circles intersecting each other...-2n9wzvb.jpg  
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    hi snighda,

    the attachment shows a general case.
    Attached Thumbnails Attached Thumbnails Two circles intersecting each other...-chord1.jpg   Two circles intersecting each other...-chord2.jpg  
    Last edited by Archie Meade; February 25th 2010 at 12:30 PM.
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    Quote Originally Posted by snigdha View Post
    Two circles intersect each other at A and B as shown in the figure. The common chord AB is produced to meet a common tangent PQ to the circles at D. Prove that DP=DQ.
    It follows from the alternate segment theorem that the triangles PDB and ADP are similar. Deduce that PD^2 = AD*BD. (In fact, this is a well-known theorem: from a point D, draw a tangent DP to a circle, and a chord DBA cutting the circle; then the square of PD is equal to the product of DA and DB.)

    Now do the same for the other circle, to see that QD^2 is also equal to AD*BD. Conclusion: PD = QD.
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