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13485Re: [EMHL] Mittenpunkt And Concurrency Of Three Euler Lines

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  • Quang Tuan Bui
    Jul 1, 2006
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      Dear Antreas,
      Very nice idea! I try only some special case and we should do together to get some general results.
      I am trying with first case: Q is fixed, and line is OH. I see the locus may be nice: one line?
      The mittenpunkt may be is one very special case: Q = incenter I and the locus is only one point: mittenpunkt or some points?
      These conjectures may be the good reasons for our efforts.
      Best regards,
      Bui Quang Tuan
      PS: Please note that these only conjectures. I can confirm only with my first mittenpunkt message.

      Antreas P. Hatzipolakis <xpolakis@...> wrote:
      On 1-07-06, Quang Tuan Bui wrote (partly):

      > Given triangle ABC, mittenpunkt Mp, incenter I. One line passing
      > through Mp perpendicular to IA cuts lines AB, AC at Ab, Ac respectively.
      > Similarly define Bc, Ba, Ca, Cb.
      > 4. The most interesting:
      > Three Euler lines of triangles MpBcCb, MpCaAc, MpAbBa are concurrent
      > at one point P.

      Dear Tuan

      We have here two interesting locus families

      Let ABC be a triangle, and Q,P two points.
      The perpendicular to QA through P intersects AB,AC at Ab,Ac, resp.
      Similarly Bc,Ba, and Ca,Cb.

      Which is the locus 1. of P 2. of Q
      such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
      are concurrent.

      Case 1 : Q = Fixed Point, P = Variable Point

      Case 2 : Q = Variable Point, P = Fixed Point

      Special Cases : P, or Q = I,O,H,G,K, etc

      And of course we can ask for concurrence of other than OH lines


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