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Math Help - orthic triangle

  1. #1
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    orthic triangle

    Let E,F,G be the excenters of triangle ABC. How can I prove that the orthic triangle (triangle with vertices at the feet of the altitudes of the bigger triangle) of EFG is ABC?

    Thanks for any help.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let E be the center af the excircle wich is tangent to BC. Then AE is the bisector of angle A. AF and AG are bisectors of the two exterior, opposite, angles of A, so F, A, G are colinear.
    Now, \widehat{BAE}=\widehat{EAC} and \widehat{FAC}=\widehat{GAB}. Then EA\perp FG.
    In the same way, FB\perp GE, \ GC\perp EF
    So ABC is the orthic triangle.
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