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Math Help - Need help on Circle properties

  1. #1
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    Need help on Circle properties


    In the diagram, the circle ABC passes through the centre X of circle ABD, and CBD is a straight line. Prove that triangle ACD is isosceles.
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  2. #2
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    Hello, acc100jt!

    I think I've got it . . .




    In the diagram, the circle ABC passes through the centre X of circle ABD,
    and CBD is a straight line.
    Prove that triangle ACD is isosceles.

    Draw radii XA, XB, and XD.

    ∆XAD is isosceles: ./XAD = /XDA = α

    ∆XBD is isoscles: ./XDB = /XBD = β . . /CBX .= .180 - B

    Hence: ./D .= ./XDA + /XDB .= .α + β


    CAXB is a cyclic quadrilateral; its opposite angles are supplementary.
    Since /CBX = 180 - β, then /CAX = β.

    Hence: ./A .= ./XAD + /XAC .= .α + β


    Therefore: ./A = /D, .CA = CD . . ∆ACD is isosceles.

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