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22173Radical axes - orthologic triangles

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  • Antreas Hatzipolakis
    Apr 22, 2014
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      Let be ABC a scalene triangle with circumcircle Gamma and let D,E,F  be the points where its incircle meets BC, AC, AB  respectively. Let the circumcircles of AEF, BFD and CDE meet Gamma a second time at X,Y,Z  respectively. Prove that the perpendiculars from A.B,C to AX, BY, CZ respectively are concurrent.

      Proposed by Michael Kural.



       Translation into triangle geometry language :-)

      Let ABC be a triangle and A'B'C' the pedal triangle of I.

      Denote:

      R1 = the radical axis of the circumcircles of  ABC and AB'C'
      R2 = the radical axis of the circumcircles of ABC and BC'A'
      R3 = the radical axis of the circumcircles of ABC and CA'B'

      The triangles ABC, bounded by (R1,R2,R3) are orthologic.

      They are orthologic for any point P (not only for I)

      Antreas