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Thread: Is a quadrilateral cyclic?

  1. #1
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    Is a quadrilateral cyclic?

    Prove that quadrilateral ABCD is cyclic if diagonal AC is perpendicular to side BC and diagonal BD is perpendicular to side AD.

    Need a hint to start...I'm stuck with this...

    Thanks in advance!
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  2. #2
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    Hello, logan6!

    Prove that quadrilateral ABCD is cyclic
    if diagonal $\displaystyle AC \perp BC$ and diagonal $\displaystyle BD \perp AD.$
    Code:
              D    *     *    C
               o- - - - - - -o
            * /   *       *   \ *
           * /        *        \ *
            /     *       *     \
          */  *               *  \*
        A o - - - - - - - - - - - o B

    $\displaystyle AC \perp BC$
    Then right triangle $\displaystyle ACB$ is inscribed in semicircle $\displaystyle ADCB.$

    $\displaystyle BD \perp AD$
    Then right triangle $\displaystyle ADB$ is inscribed in semicircle $\displaystyle ADC B.$


    Hence, quadrilateral $\displaystyle ABCD$ is inscribed in semicircle $\displaystyle ADCB.$


    Therefore: .quadrilateral $\displaystyle ABCD$ is cyclic.

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  3. #3
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    Thank you very much! You made my bulb glow!

    Note: I made a petite modification. I hope you don't mind.


    Then right triangle is inscribed in circle ACB. (r=AB/2, center in the middle of hypotenuse AB)


    Then right triangle is inscribed in circle ADB.
    (r=AB/2, center in the middle of hypotenuse AB))


    Hence, quadrilateral is inscribed in circle


    Therefore: .quadrilateral is cyclic.
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