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Math Help - Proving two bisectors are equal

  1. #1
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    Proving two bisectors are equal

    In a triangle ABC, angle B is 60 degrees. Angles A and C are bisected to points D and E respectively, on the edge of the triangle (see diagram). They cross at point O. Prove that OD = OE



    I've gone through it a little and got this.

    2x + 2y + 60 = 180
    2x + 2y = 120
    x + y = 60

    I don't know if this helps at all.

    Remember, angles A and C are bisected, so the two adjacent angles are equal to each other (x, x and y, y)

    Thanks for any help, BG
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  2. #2
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    Geometry Proof

    Hello BG5965
    Quote Originally Posted by BG5965 View Post
    In a triangle ABC, angle B is 60 degrees. Angles A and C are bisected to points D and E respectively, on the edge of the triangle (see diagram). They cross at point O. Prove that OD = OE



    I've gone through it a little and got this.

    2x + 2y + 60 = 180
    2x + 2y = 120
    x + y = 60

    I don't know if this helps at all.

    Remember, angles A and C are bisected, so the two adjacent angles are equal to each other (x, x and y, y)

    Thanks for any help, BG
    Join OB, and note that it is the bisector of \angle B, since the angle bisectors of a triangle are concurrent.

    \Rightarrow \angle OBD = \angle OBE = 30^o

    Also \angle EOA = x + y (exterior angle of \triangle AOC)

    \Rightarrow EOA = 60^o = \angle EBD

    \Rightarrow EODB is a cyclic quadrilateral (exterior angle = interior opposite angle)

    Now join ED, and use angles in the same segment:

    \angle OBD = \angle OED = 30^o and \angle OBE = \angle ODE = 30^o

    \Rightarrow \angle ODE = \angle OED

    \Rightarrow EO = OD (isosceles \triangle OED)

    Grandad
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