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Math Help - Geometry help.

  1. #1
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    Geometry help.

    Extend the bisectors of angles A, B, and C of triangle ABC to meet the circumcircle at points X, Y, and Z. Show that I is the orthocenter of triangle ABC.
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  2. #2
    MHF Contributor red_dog's Avatar
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    I think I is the orthocenter of XYZ. (not ABC)

    Let AX\cap YZ=M

    m(\widehat{MIZ})=\frac{m(arcAZ)+m(arcCX)}{2}=\frac  {m(arcAB)+m(arcBC)}{4}=\frac{m(arcABC)}{4}=

    =\frac{360^{\circ}-m(arcAC)}{4}=90^{\circ}-\frac{m(arcAC)}{4}=90^{\circ}-\frac{m(arcYC)}{2}=90^{\circ}-m(\widehat{MZI})

    Then, m(\widehat{MIZ})+m(\widehat{MZI})=90^{\circ}\Right  arrow m(\widehat{ZMI})=90^{\circ}\Rightarrow AX\perp YZ

    In the same way BY\perp XZ, \ CZ\perp YX
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