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Re: excircle and circumcircle

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  • Darij Grinberg
    Dear Alexey, ... You can add the following equivalent condition: 5. If I is the incenter of triangle ABC, and Hb and Hc are the feet of the altitudes from B
    Message 1 of 2 , Jan 27, 2004
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      Dear Alexey,

      In Hyacinthos message #9139, you wrote:

      >> Let AX and BY are the bisectrix of triangle ABC,
      >> O, R - the center and radius of its circumcircle,
      >> I1, r1 - the center and radius of excircle
      >> touching the sideline AB, A', B', C' - the
      >> touching points of excircle with BC, CA, AB.
      >> Then next conditions are equivalent:
      >> 1. R=r1.
      >> 2. O is in line XY.
      >> 3. O is the orthocenter of A'B'C' (I suppose
      >> also that A'B'C' is autopolar with respect to
      >> circumcicle of ABC).
      >> 4. cosA+cosB=cosC.

      You can add the following equivalent condition:
      5. If I is the incenter of triangle ABC, and Hb
      and Hc are the feet of the altitudes from B and C,
      then I lies on HbHc.

      The equivalence of the conditions 2. and 5. was the
      subject of Problem 4 in the Bundeswettbewerb
      Mathematik (German National Mathematics Olympiad)
      2002, 2 round. In fact, it was a very hard problem;
      I was lucky enough to know the method of trilinear
      coordinates to solve it!

      Sincerely,
      Darij Grinberg
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