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Math Help - Triangle problem

  1. #1
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    Triangle problem

    Acute triangle is given. The angle at point C is 45 . Points A1 and B1 are the feet of the altitudes from points A and B, respectively . Let point H be an orthocenter of a triangle. On segments AA1 and BC and are points D and E, respectively, so that A1D=A1E=A1B1 . Prove that:

    1. CH=DE

    2. A1B1^2=(A1B^2+A1C^2)/2
    Last edited by matt1; June 24th 2013 at 11:21 AM.
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  2. #2
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    Re: Triangle problem

    Hi,
    I think this problem has a fairly easy solution using analytic geometry. I've attached a diagram where the problem has been coordinatized; the diagram shows that the assumption that all angles of the triangle are acute is not necessary. I left most of the solution to you, but if you have question, just reply to this posting.

    Triangle problem-mhfgeometry29.png
    Last edited by johng; June 25th 2013 at 06:05 AM.
    Thanks from matt1
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  3. #3
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    Re: Triangle problem

    Thanks for the help johng ! But i was wondering how did you get coordinates of B1 (a+b/2, a-b/2) ?
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  4. #4
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    Re: Triangle problem

    Hi Matt1,
    I've attached a slightly modified figure which shows how the coordinates for B1 are found. Again, reply if you have further questions.

    Triangle problem-mhfgeometry29a.png
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  5. #5
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    Re: Triangle problem

    Hi johng, again thanks for your fast reply and help, it's all clear now !
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