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Math Help - 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 AC \perp BC and diagonal  BD \perp AD.
    Code:
              D    *     *    C
               o- - - - - - -o
            * /   *       *   \ *
           * /        *        \ *
            /     *       *     \
          */  *               *  \*
        A o - - - - - - - - - - - o B

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

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


    Hence, quadrilateral ABCD is inscribed in semicircle  ADCB.


    Therefore: .quadrilateral  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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