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Math Help - circle geometry

  1. #1
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    circle geometry

    In the figure , two circles intersect at P and Q , and AQ is the diameter of circle APQC . THe lines APD , AQB, and DQC are straight . Prove taht

     <br />
\angle ADC=\angle ABC <br />
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  2. #2
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    Quote Originally Posted by thereddevils View Post
    In the figure , two circles intersect at P and Q , and AQ is the diameter of circle APQC . THe lines APD , AQB, and DQC are straight . Prove taht

     <br />
\angle ADC=\angle ABC <br />
    since AQ is a diameter, you have angle ACQ = 90 degrees
    also, angle APQ = 90 degrees
    let angles AQC = DQB = a (vertically opposite angles)
    so you have angle CAQ = 90^o - a
    since APD is straight line,
    angles DPQ = APQ = 90 degrees
    so you can prove that QD is diameter of circle PQBD
    Therefore, angle QBD = 90 degrees

    now you can prove that triangles ACQ and DBQ are similar,
    with angles CAQ = BDQ = 90^o - a
    Hence, ADBC is an inscribed quadrilateral, and so angles ADC = ABC
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