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Math Help - geometry - pentagons

  1. #1
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    geometry - pentagons

    In a pentagon ABCDE,

    AB = AE

    BC = ED

    BD = EC

    Prove that AC = AD
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Hint:
    Block your pentagon with a circle.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Hint:
    Block your pentagon with a circle.
    My mistake! Sorry!
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  4. #4
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    Quote Originally Posted by the undertaker View Post
    In a pentagon ABCDE,

    AB = AE

    BC = ED

    BD = EC

    Prove that AC = AD
    (1) Triangle EDC is congruent to triangle BCD (SSS).

    (2) Since angles in the same segment are equal(angle DEC = angle DBC), we can deduce that this pentagon is inscribed in a circle.

    (3) Chords AB=AE implies arc AB=AE , angle,x subtended by these arcs are the same. Similarly, BC=ED, angle,y subtended will be the same too.

    angle ADC=angle ACD= x+y and since base angles are equal, triangle ADC is an isosceles triangle.
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