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LOCUS

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  • Antreas
    Let ABC be a triangle and A B C , A B C the cevian, pedal triangles of P, resp. Denote: Ab, Ac = the reflections of A in BB , CC Bc, Ba = the reflections of
    Message 1 of 26 , Feb 25, 2013
      Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal
      triangles of P, resp.

      Denote:

      Ab, Ac = the reflections of A' in BB', CC'

      Bc, Ba = the reflections of B' in CC', AA'

      Ca, Cb = the reflections of C' in AA', BB'

      Ea, Eb, Ec = the Euler Lines of A'AbAc, B'BcBa, C'CaCb, resp. (concurrent at P = common circumcenter of the triangles)

      La, Lb, Lc = the parallels through A",B",C" to Ea,Eb,Ec, resp.

      For P = I, the lines La,Lb,Lc concur on the pedal circle of I.
      (antipode of Feuerbach point)

      For P = H (A'B'C' = A"B"C") the lines La,Lb,Lc concur
      on the pedal circle of H (=NPC).

      Which is the point of concurrence?

      In general:
      Which is the locus of P such that the lines La,Lb,Lc
      are concurrent?

      Antreas
    • Angel
      Dear Antreas, If P=I the lines La,Lb,Lc intersect at X(1317) is the antipode of Feuerbach point on the incircle. If P=H the lines La,Lb,Lc concur in X(1986)=
      Message 2 of 26 , Feb 26, 2013
        Dear Antreas,

        If P=I the lines La,Lb,Lc intersect at X(1317) is the antipode of Feuerbach point on the incircle.

        If P=H the lines La,Lb,Lc concur in X(1986)= HATZIPOLAKIS REFLECTION POINT (Antreas Hatzipolakis,Hyacinthos 7868,9/12/03;coordinates by Barry Wolk,Hyacinthos 7876,9/13/03)

        X(1986)=(a^2(4SA^2-b^2c^2)(a^2(SA^2-SB SC)-SA(c^2-b^2)^2)/SA: ... : ...)


        In general:
        The lines La,Lb,Lc are concurrent if P is on the algebraic curve (Gamma) of degree nine (SA, SB, SC usual Conway notation):


        (SA+SB)^3(SA+SC)(SA*SB-2SA*SC+SB*SC)x^6y^3-

        (SA+SB)^3(SA+SC)(5SA*SB+5SA*SC-4SB*SC)x^5y^4+

        (SA+SB)^3(SB+SC)(5SA*SB-4SA*SC+5SB*SC)x^4y^5-

        (SA+SB)^3(SB+SC)(SA*SB+SA*SC-2SB*SC)x^3y^6+

        (SA+SB)^2x^2y^2z((-SA-SC)(-4SA^2SB+SA(5SA-3SB)SC+(SA+SB)SC^2)x^4-
        2(SA+SC)(SA(SB-SC)^2+5SA^2(SB+SC)+SB*SC(SB+SC))x^3y-
        4(SA-SB)SC(SB*SC+SA(SB+SC))x^2y^2+
        2(SB+SC)(SA^2(SB+SC)+SB*SC(5SB+SC)+SA(5SB^2-2SB*SC+SC^2))x*y^3+
        (SB+SC)(-SA(4SB-SC)(SB+SC)+SB*SC(5SB+SC))y^4)+

        (SA+SB)x*y*z^2((SA+SC)^2(SA*SB(5SA+SB)-(4SA-SB)(SA+SB)SC)x^5-
        (SA+SC)(SA^2(19SB-14SC)(SB+SC)+SB^2SC(7SB+11*SC)+
        SA*SB(7SB^2-8SB*SC-3SC^2))x^3y^2+(SB+SC)*
        (SA^2SB(7SA+19SB)+SA(7SA^2-8SA*SB+5SB^2)SC+
        (11SA-14SB)(SA+SB)SC^2)x^2y^3-(SB+SC)^2(SA*SB(SA+5SB)+
        (SA-4SB)(SA+SB)SC)y^5)+

        (SA+SB)z^3((-(SA+SC)^3)(-2SA*SB+
        (SA+SB)SC)x^6+2(SA+SC)^2(SA(SB-SC)^2+5SA^2(SB+SC)+
        SB*SC(SB+SC))x^5y-(SA+SC)(SA^2(14SB-19SC)*

        (SB+SC)-SB*SC^2(11SB+7SC)+SA*SC(3SB^2+8SB*SC-7SC^2))*
        x^4y^2+(SB+SC)(SA^2(14SB-11SC)(SB+SC)-
        SB*SC^2(19SB+7SC)+SA*SC(-5SB^2+8SB*SC-7SC^2))*
        x^2y^4-2(SB+SC)^2(SA^2(SB+SC)+SB*SC(5SB+SC)+
        SA(5SB^2-2SB*SC+SC^2))x*y^5+(SB+SC)^3(-2SA*SB+(SA+SB)SC)*y^6)+

        ((SA+SB)(SA+SC)^3(-4SB*SC+5SA(SB+SC))x^5+

        4SB(SA-SC)(SA+SC)^2(SB*SC+SA(SB+SC))x^4y-

        (SA+SC)(SB+SC)(SA^2SB(7SA+11SB)+
        SA(7SA^2-8SA*SB-3SB^2)SC+(19SA-14SB)(SA+SB)SC^2)x^3y^2+

        (SA+SC)(SB+SC)(SA^2(11SB-14SC)(SB+SC)+
        SB^2SC(7SB+19SC)+SA*SB(7SB^2-8SB*SC+5SC^2))x^2y^3-

        4SA(SB-SC)(SB+SC)^2(SB*SC+SA(SB+SC))x*y^4-

        (SA+SB)(SB+SC)^3(5SA*SB-4SA*SC+5SB*SC)y^5)z^4-

        (SA+SC)(SB+SC)((SA+SC)x^2-(SB+SC)y^2)((SA+SC)(-4SA*SB+5(SA+SB)SC)x^2+
        2(SA*SB(SA+SB)+(SA-SB)^2SC+5(SA+SB)SC^2)x*y+
        (SB+SC)(-4SA*SB+5(SA+SB)SC)y^2)z^5+

        (SA+SC)(SB+SC)((SA+SC)(-2SB*SC+SA(SB+SC))x-(SB+SC)(SA(SB-2SC)+SB*SC)y)*(SA*x^2+SB*y^2+SC(x+y)^2)z^6 =0


        (There must be a better simplification of this equation!)


        This curve (Gamma) contains points isodynamic (X13, X14) but the corresponding triangles AAbAc, BBcBa and CCaCb are equilateral.

        The triangle center X(74) -isogonal conjugate of Euler infinity point- is on the curve (Gamma) and the Euler Lines of A'AbAc, B'BcBa, C'CaCb
        passing through A'', B'', C'' resp. (concurrent at X74 = common circumcenter of the triangles).


        Best regards
        Angel Montesdeoca

        --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
        >
        > Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal
        > triangles of P, resp.
        >
        > Denote:
        >
        > Ab, Ac = the reflections of A' in BB', CC'
        >
        > Bc, Ba = the reflections of B' in CC', AA'
        >
        > Ca, Cb = the reflections of C' in AA', BB'
        >
        > Ea, Eb, Ec = the Euler Lines of A'AbAc, B'BcBa, C'CaCb, resp. (concurrent at P = common circumcenter of the triangles)
        >
        > La, Lb, Lc = the parallels through A",B",C" to Ea,Eb,Ec, resp.
        >
        > For P = I, the lines La,Lb,Lc concur on the pedal circle of I.
        > (antipode of Feuerbach point)
        >
        > For P = H (A'B'C' = A"B"C") the lines La,Lb,Lc concur
        > on the pedal circle of H (=NPC).
        >
        > Which is the point of concurrence?
        >
        > In general:
        > Which is the locus of P such that the lines La,Lb,Lc
        > are concurrent?
        >
        > Antreas
        >
      • Angel
        More information on the algebraic curve of degree nine (Gamma): - Passes through the vertices of the triangle ABC. - The vertices are multiple points of order
        Message 3 of 26 , Feb 26, 2013
          More information on the algebraic curve of degree nine (Gamma):

          - Passes through the vertices of the triangle ABC.

          - The vertices are multiple points of order 3.

          - The real tangents at the vertices of ABC intersect at the point X(74).


          Angel M.

          --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@...> wrote:
          >
          >
          > Dear Antreas,
          >
          > If P=I the lines La,Lb,Lc intersect at X(1317) is the antipode of Feuerbach point on the incircle.
          >
          > If P=H the lines La,Lb,Lc concur in X(1986)= HATZIPOLAKIS REFLECTION POINT (Antreas Hatzipolakis,Hyacinthos 7868,9/12/03;coordinates by Barry Wolk,Hyacinthos 7876,9/13/03)
          >
          > X(1986)=(a^2(4SA^2-b^2c^2)(a^2(SA^2-SB SC)-SA(c^2-b^2)^2)/SA: ... : ...)
          >
          >
          > In general:
          > The lines La,Lb,Lc are concurrent if P is on the algebraic curve (Gamma) of degree nine (SA, SB, SC usual Conway notation):
          >
          >
          > (SA+SB)^3(SA+SC)(SA*SB-2SA*SC+SB*SC)x^6y^3-
          >
          > (SA+SB)^3(SA+SC)(5SA*SB+5SA*SC-4SB*SC)x^5y^4+
          >
          > (SA+SB)^3(SB+SC)(5SA*SB-4SA*SC+5SB*SC)x^4y^5-
          >
          > (SA+SB)^3(SB+SC)(SA*SB+SA*SC-2SB*SC)x^3y^6+
          >
          > (SA+SB)^2x^2y^2z((-SA-SC)(-4SA^2SB+SA(5SA-3SB)SC+(SA+SB)SC^2)x^4-
          > 2(SA+SC)(SA(SB-SC)^2+5SA^2(SB+SC)+SB*SC(SB+SC))x^3y-
          > 4(SA-SB)SC(SB*SC+SA(SB+SC))x^2y^2+
          > 2(SB+SC)(SA^2(SB+SC)+SB*SC(5SB+SC)+SA(5SB^2-2SB*SC+SC^2))x*y^3+
          > (SB+SC)(-SA(4SB-SC)(SB+SC)+SB*SC(5SB+SC))y^4)+
          >
          > (SA+SB)x*y*z^2((SA+SC)^2(SA*SB(5SA+SB)-(4SA-SB)(SA+SB)SC)x^5-
          > (SA+SC)(SA^2(19SB-14SC)(SB+SC)+SB^2SC(7SB+11*SC)+
          > SA*SB(7SB^2-8SB*SC-3SC^2))x^3y^2+(SB+SC)*
          > (SA^2SB(7SA+19SB)+SA(7SA^2-8SA*SB+5SB^2)SC+
          > (11SA-14SB)(SA+SB)SC^2)x^2y^3-(SB+SC)^2(SA*SB(SA+5SB)+
          > (SA-4SB)(SA+SB)SC)y^5)+
          >
          > (SA+SB)z^3((-(SA+SC)^3)(-2SA*SB+
          > (SA+SB)SC)x^6+2(SA+SC)^2(SA(SB-SC)^2+5SA^2(SB+SC)+
          > SB*SC(SB+SC))x^5y-(SA+SC)(SA^2(14SB-19SC)*
          >
          > (SB+SC)-SB*SC^2(11SB+7SC)+SA*SC(3SB^2+8SB*SC-7SC^2))*
          > x^4y^2+(SB+SC)(SA^2(14SB-11SC)(SB+SC)-
          > SB*SC^2(19SB+7SC)+SA*SC(-5SB^2+8SB*SC-7SC^2))*
          > x^2y^4-2(SB+SC)^2(SA^2(SB+SC)+SB*SC(5SB+SC)+
          > SA(5SB^2-2SB*SC+SC^2))x*y^5+(SB+SC)^3(-2SA*SB+(SA+SB)SC)*y^6)+
          >
          > ((SA+SB)(SA+SC)^3(-4SB*SC+5SA(SB+SC))x^5+
          >
          > 4SB(SA-SC)(SA+SC)^2(SB*SC+SA(SB+SC))x^4y-
          >
          > (SA+SC)(SB+SC)(SA^2SB(7SA+11SB)+
          > SA(7SA^2-8SA*SB-3SB^2)SC+(19SA-14SB)(SA+SB)SC^2)x^3y^2+
          >
          > (SA+SC)(SB+SC)(SA^2(11SB-14SC)(SB+SC)+
          > SB^2SC(7SB+19SC)+SA*SB(7SB^2-8SB*SC+5SC^2))x^2y^3-
          >
          > 4SA(SB-SC)(SB+SC)^2(SB*SC+SA(SB+SC))x*y^4-
          >
          > (SA+SB)(SB+SC)^3(5SA*SB-4SA*SC+5SB*SC)y^5)z^4-
          >
          > (SA+SC)(SB+SC)((SA+SC)x^2-(SB+SC)y^2)((SA+SC)(-4SA*SB+5(SA+SB)SC)x^2+
          > 2(SA*SB(SA+SB)+(SA-SB)^2SC+5(SA+SB)SC^2)x*y+
          > (SB+SC)(-4SA*SB+5(SA+SB)SC)y^2)z^5+
          >
          > (SA+SC)(SB+SC)((SA+SC)(-2SB*SC+SA(SB+SC))x-(SB+SC)(SA(SB-2SC)+SB*SC)y)*(SA*x^2+SB*y^2+SC(x+y)^2)z^6 =0
          >
          >
          > (There must be a better simplification of this equation!)
          >
          >
          > This curve (Gamma) contains points isodynamic (X13, X14) but the corresponding triangles AAbAc, BBcBa and CCaCb are equilateral.
          >
          > The triangle center X(74) -isogonal conjugate of Euler infinity point- is on the curve (Gamma) and the Euler Lines of A'AbAc, B'BcBa, C'CaCb
          > passing through A'', B'', C'' resp. (concurrent at X74 = common circumcenter of the triangles).
          >
          >
          > Best regards
          > Angel Montesdeoca
          >
          > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:
          > >
          > > Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal
          > > triangles of P, resp.
          > >
          > > Denote:
          > >
          > > Ab, Ac = the reflections of A' in BB', CC'
          > >
          > > Bc, Ba = the reflections of B' in CC', AA'
          > >
          > > Ca, Cb = the reflections of C' in AA', BB'
          > >
          > > Ea, Eb, Ec = the Euler Lines of A'AbAc, B'BcBa, C'CaCb, resp. (concurrent at P = common circumcenter of the triangles)
          > >
          > > La, Lb, Lc = the parallels through A",B",C" to Ea,Eb,Ec, resp.
          > >
          > > For P = I, the lines La,Lb,Lc concur on the pedal circle of I.
          > > (antipode of Feuerbach point)
          > >
          > > For P = H (A'B'C' = A"B"C") the lines La,Lb,Lc concur
          > > on the pedal circle of H (=NPC).
          > >
          > > Which is the point of concurrence?
          > >
          > > In general:
          > > Which is the locus of P such that the lines La,Lb,Lc
          > > are concurrent?
          > >
          > > Antreas
          > >
          >
        • Antreas Hatzipolakis
          Dear Angel Thank you. I came to this configuration trying to find three homocentric (concentric) circles but not by construction (ie not by taking a point as
          Message 4 of 26 , Feb 26, 2013
            Dear Angel

            Thank you.

            I came to this configuration trying to find three homocentric (concentric)
            circles
            but not by construction (ie not by taking a point as center).

            Antreas



            On Tue, Feb 26, 2013 at 2:57 PM, Angel <amontes1949@...> wrote:

            > **
            >
            >
            >
            > Dear Antreas,
            >
            > If P=I the lines La,Lb,Lc intersect at X(1317) is the antipode of
            > Feuerbach point on the incircle.
            >
            > If P=H the lines La,Lb,Lc concur in X(1986)= HATZIPOLAKIS REFLECTION POINT
            > (Antreas Hatzipolakis,Hyacinthos 7868,9/12/03;coordinates by Barry
            > Wolk,Hyacinthos 7876,9/13/03)
            >
            > X(1986)=(a^2(4SA^2-b^2c^2)(a^2(SA^2-SB SC)-SA(c^2-b^2)^2)/SA: ... : ...)
            >
            > In general:
            > The lines La,Lb,Lc are concurrent if P is on the algebraic curve (Gamma)
            > of degree nine (SA, SB, SC usual Conway notation):
            >
            > (SA+SB)^3(SA+SC)(SA*SB-2SA*SC+SB*SC)x^6y^3-
            >
            > (SA+SB)^3(SA+SC)(5SA*SB+5SA*SC-4SB*SC)x^5y^4+
            >
            > (SA+SB)^3(SB+SC)(5SA*SB-4SA*SC+5SB*SC)x^4y^5-
            >
            > (SA+SB)^3(SB+SC)(SA*SB+SA*SC-2SB*SC)x^3y^6+
            >
            > (SA+SB)^2x^2y^2z((-SA-SC)(-4SA^2SB+SA(5SA-3SB)SC+(SA+SB)SC^2)x^4-
            > 2(SA+SC)(SA(SB-SC)^2+5SA^2(SB+SC)+SB*SC(SB+SC))x^3y-
            > 4(SA-SB)SC(SB*SC+SA(SB+SC))x^2y^2+
            > 2(SB+SC)(SA^2(SB+SC)+SB*SC(5SB+SC)+SA(5SB^2-2SB*SC+SC^2))x*y^3+
            > (SB+SC)(-SA(4SB-SC)(SB+SC)+SB*SC(5SB+SC))y^4)+
            >
            > (SA+SB)x*y*z^2((SA+SC)^2(SA*SB(5SA+SB)-(4SA-SB)(SA+SB)SC)x^5-
            > (SA+SC)(SA^2(19SB-14SC)(SB+SC)+SB^2SC(7SB+11*SC)+
            > SA*SB(7SB^2-8SB*SC-3SC^2))x^3y^2+(SB+SC)*
            > (SA^2SB(7SA+19SB)+SA(7SA^2-8SA*SB+5SB^2)SC+
            > (11SA-14SB)(SA+SB)SC^2)x^2y^3-(SB+SC)^2(SA*SB(SA+5SB)+
            > (SA-4SB)(SA+SB)SC)y^5)+
            >
            > (SA+SB)z^3((-(SA+SC)^3)(-2SA*SB+
            > (SA+SB)SC)x^6+2(SA+SC)^2(SA(SB-SC)^2+5SA^2(SB+SC)+
            > SB*SC(SB+SC))x^5y-(SA+SC)(SA^2(14SB-19SC)*
            >
            > (SB+SC)-SB*SC^2(11SB+7SC)+SA*SC(3SB^2+8SB*SC-7SC^2))*
            > x^4y^2+(SB+SC)(SA^2(14SB-11SC)(SB+SC)-
            > SB*SC^2(19SB+7SC)+SA*SC(-5SB^2+8SB*SC-7SC^2))*
            > x^2y^4-2(SB+SC)^2(SA^2(SB+SC)+SB*SC(5SB+SC)+
            > SA(5SB^2-2SB*SC+SC^2))x*y^5+(SB+SC)^3(-2SA*SB+(SA+SB)SC)*y^6)+
            >
            > ((SA+SB)(SA+SC)^3(-4SB*SC+5SA(SB+SC))x^5+
            >
            > 4SB(SA-SC)(SA+SC)^2(SB*SC+SA(SB+SC))x^4y-
            >
            > (SA+SC)(SB+SC)(SA^2SB(7SA+11SB)+
            > SA(7SA^2-8SA*SB-3SB^2)SC+(19SA-14SB)(SA+SB)SC^2)x^3y^2+
            >
            > (SA+SC)(SB+SC)(SA^2(11SB-14SC)(SB+SC)+
            > SB^2SC(7SB+19SC)+SA*SB(7SB^2-8SB*SC+5SC^2))x^2y^3-
            >
            > 4SA(SB-SC)(SB+SC)^2(SB*SC+SA(SB+SC))x*y^4-
            >
            > (SA+SB)(SB+SC)^3(5SA*SB-4SA*SC+5SB*SC)y^5)z^4-
            >
            > (SA+SC)(SB+SC)((SA+SC)x^2-(SB+SC)y^2)((SA+SC)(-4SA*SB+5(SA+SB)SC)x^2+
            > 2(SA*SB(SA+SB)+(SA-SB)^2SC+5(SA+SB)SC^2)x*y+
            > (SB+SC)(-4SA*SB+5(SA+SB)SC)y^2)z^5+
            >
            > (SA+SC)(SB+SC)((SA+SC)(-2SB*SC+SA(SB+SC))x-(SB+SC)(SA(SB-2SC)+SB*SC)y)*(SA*x^2+SB*y^2+SC(x+y)^2)z^6
            > =0
            >
            > (There must be a better simplification of this equation!)
            >
            > This curve (Gamma) contains points isodynamic (X13, X14) but the
            > corresponding triangles AAbAc, BBcBa and CCaCb are equilateral.
            >
            > The triangle center X(74) -isogonal conjugate of Euler infinity point- is
            > on the curve (Gamma) and the Euler Lines of A'AbAc, B'BcBa, C'CaCb
            > passing through A'', B'', C'' resp. (concurrent at X74 = common
            > circumcenter of the triangles).
            >
            > Best regards
            > Angel Montesdeoca
            >
            >
            > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
            > >
            > > Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal
            > > triangles of P, resp.
            > >
            > > Denote:
            > >
            > > Ab, Ac = the reflections of A' in BB', CC'
            > >
            > > Bc, Ba = the reflections of B' in CC', AA'
            > >
            > > Ca, Cb = the reflections of C' in AA', BB'
            > >
            > > Ea, Eb, Ec = the Euler Lines of A'AbAc, B'BcBa, C'CaCb, resp.
            > (concurrent at P = common circumcenter of the triangles)
            > >
            > > La, Lb, Lc = the parallels through A",B",C" to Ea,Eb,Ec, resp.
            > >
            > > For P = I, the lines La,Lb,Lc concur on the pedal circle of I.
            > > (antipode of Feuerbach point)
            > >
            > > For P = H (A'B'C' = A"B"C") the lines La,Lb,Lc concur
            > > on the pedal circle of H (=NPC).
            > >
            > > Which is the point of concurrence?
            > >
            > > In general:
            > > Which is the locus of P such that the lines La,Lb,Lc
            > > are concurrent?
            > >
            > > Antreas
            > >
            >
            >
            >



            --
            http://anopolis72000.blogspot.com/


            [Non-text portions of this message have been removed]
          • Antreas Hatzipolakis
            [APH] ... More for P = H: The NPCs of A AbAc, B BcBa, C CaCb are concurrent on the NPC of A B C (On the Poncelet point of H with respect A B C ie the point
            Message 5 of 26 , Feb 26, 2013
              [APH]


              >
              > Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal
              > triangles of P, resp.
              >
              > Denote:
              >
              > Ab, Ac = the reflections of A' in BB', CC'
              >
              > Bc, Ba = the reflections of B' in CC', AA'
              >
              > Ca, Cb = the reflections of C' in AA', BB'
              >
              > Ea, Eb, Ec = the Euler Lines of A'AbAc, B'BcBa, C'CaCb, resp. (concurrent
              > at P = common circumcenter of the triangles)
              >
              > La, Lb, Lc = the parallels through A",B",C" to Ea,Eb,Ec, resp.
              >
              > For P = I, the lines La,Lb,Lc concur on the pedal circle of I.
              > (antipode of Feuerbach point)
              >
              > For P = H (A'B'C' = A"B"C") the lines La,Lb,Lc concur
              > on the pedal circle of H (=NPC).
              >
              >
              >
              More for P = H:

              The NPCs of A'AbAc, B'BcBa, C'CaCb are concurrent on
              the NPC of A'B'C' (On the Poncelet point of H with respect
              A'B'C' ie the point of concurrence of the NPCs of
              A'B'C', HB'C', HC'A', HA'B'. So we have seven concurrent NPCs)

              The parallels through A,B,C to the (concurrent at H)
              Euler lines Ea, Eb, Ec of A'AbAc, B'BcBa, C'CaCb, resp. concur on the
              circumcircle of ABC on the antipode of the Euler line
              reflection point. And the perpendiculars, on the Euler line reflection
              point.


              APH


              [Non-text portions of this message have been removed]
            • Antreas Hatzipolakis
              Let ABC be a triangle and P a point. Denote: P1,P2,P3 = the P-points of PBC,PCA,PAB resp. P12,P13 = the reflections of P1 in AC,AB, resp. P23,P21 = the
              Message 6 of 26 , May 29, 2016
                Let ABC be a triangle and P a point.

                Denote:

                P1,P2,P3 = the P-points of PBC,PCA,PAB resp.

                P12,P13 = the reflections of P1 in AC,AB, resp.
                P23,P21 = the reflections of P2 in BA,BC, resp.
                P31,P32 = the reflections of P3 in CB,CA, resp.

                Which is the locus of P such that the perpendicular bisectors of

                P12P13, P23P21, P31P32 are concurrent ?

                (equivalently: Let M1,M2,M3 be the midpoints of P12P13, P23P21, P31P32, resp.

                Which is the locus of P such that ABC, M1M2M3 are perspective?)

                1. P = O

                O1,O2,O3 = the circumcenters of OBC,OCA,OAB resp.
                O12,O13 = the reflections of O1 in AC,AB, resp.
                O23,O21 = the reflections of O2 in BA,BC, resp.
                O31,O32 = the reflections of O3 in CB,CA, resp.

                The perpendicular bisectors of O12O13,O23O21,O31O32 are concurrent at N.

                2. P = I

                I1,I2,I3 = the Incenters of IBC,ICA,IAB resp.
                I12,I13 = the reflections of I1 in AC,AB, resp.
                I23,I21 = the reflections of I2 in BA,BC, resp.
                I31,I32 = the reflections of I3 in CB,CA, resp.

                The perpendicular bisectors of I12I13,I23I21,I31I32 are concurrent at ??? (on
                the OI line)

                3. P = N

                N1, N2, N3 = the NPC centers of NBC,NCA,NAB, resp.

                N12,N13 = the reflections of N1 in AC,AB, resp.
                N23,N21 = the reflections of N2 in BA,BC, resp.
                N31,N32 = the reflections of N3 in CB,CA, resp.

                The perpendicular bisectors of N12N13, N23N21, N31N32 are concurrent at ??.

                Antreas





              • xpolakis
                [APH]: 3. P = N N1, N2, N3 = the NPC centers of NBC,NCA,NAB, resp. N12,N13 = the reflections of N1 in AC,AB, resp. N23,N21 = the reflections of N2 in BA,BC,
                Message 7 of 26 , May 29, 2016


                  [APH]:

                  3. P = N

                  N1, N2, N3 = the NPC centers of NBC,NCA,NAB, resp.

                  N12,N13 = the reflections of N1 in AC,AB, resp.
                  N23,N21 = the reflections of N2 in BA,BC, resp.
                  N31,N32 = the reflections of N3 in CB,CA, resp.

                  The perpendicular bisectors of N12N13, N23N21, N31N32 are concurrent at ??.

                  Antreas

                  [Angel Montesdeoca]:


                  The perpendicular bisectors of N12N13, N23N21, N31N32 are cevians of  X(1493) = NAPOLEON CROSSSUM

                  Best regards,
                  Angel M.






                   
                • Antreas Hatzipolakis
                  [APH] Let ABC be a triangle and P a point. Denote: P1,P2,P3 = the P-points of PBC,PCA,PAB resp. P12,P13 = the reflections of P1 in AC,AB, resp. P23,P21 = the
                  Message 8 of 26 , May 30, 2016
                    [APH]

                    Let ABC be a triangle and P a point.

                    Denote:

                    P1,P2,P3 = the P-points of PBC,PCA,PAB resp.

                    P12,P13 = the reflections of P1 in AC,AB, resp.
                    P23,P21 = the reflections of P2 in BA,BC, resp.
                    P31,P32 = the reflections of P3 in CB,CA, resp.

                    Which is the locus of P such that the perpendicular bisectors of

                    P12P13, P23P21, P31P32 are concurrent ?

                    (equivalently: Let M1,M2,M3 be the midpoints of P12P13, P23P21, P31P32, resp.

                    Which is the locus of P such that ABC, M1M2M3 are perspective?)

                    1. P = O

                    O1,O2,O3 = the circumcenters of OBC,OCA,OAB resp.
                    O12,O13 = the reflections of O1 in AC,AB, resp.
                    O23,O21 = the reflections of O2 in BA,BC, resp.
                    O31,O32 = the reflections of O3 in CB,CA, resp.

                    The perpendicular bisectors of O12O13,O23O21,O31O32 are concurrent at N.

                    2. P = I

                    I1,I2,I3 = the Incenters of IBC,ICA,IAB resp.
                    I12,I13 = the reflections of I1 in AC,AB, resp.
                    I23,I21 = the reflections of I2 in BA,BC, resp.
                    I31,I32 = the reflections of I3 in CB,CA, resp.

                    The perpendicular bisectors of I12I13,I23I21,I31I32 are concurrent at ??? (on

                    the OI line)

                    3. P = N

                    N1, N2, N3 = the NPC centers of NBC,NCA,NAB, resp.

                    N12,N13 = the reflections of N1 in AC,AB, resp.
                    N23,N21 = the reflections of N2 in BA,BC, resp.
                    N31,N32 = the reflections of N3 in CB,CA, resp.

                    The perpendicular bisectors of N12N13, N23N21, N31N32 are concurrent at ??.

                    Antreas


                    [César E. Lozada]:
                     

                    Dear Antreas,

                    The list is longer.

                    The appearance of i(j) in the following list means that the perpendicular bisectors for P=X(i) concur at X(j). If no j is given it means that they concur at a non-ETC center:

                    1(1129), 3(5), 5(1493), 6(141), 13(15), 15(17), 17(61), 20(3), 31, 32, 75, 76, 140, 176, 365, 366, 376(4550), 381, 382, 399(5671), 485(371), 546, 547, 548, 549, 550, 560, 561, 631, 632, 1501, 1502, 1656, 1657, 1917, 1928, 2042, 2043, 2044, 2045, 2046, 2671, 2675, 2676

                    Note: Found with i<=3054.

                    Regards

                    César
                  • Antreas Hatzipolakis
                    ... Dear Angel and Cesar, Thank you for your responses. Now, I think the same is true if we replace the Incenters of the component triangles with excenters.
                    Message 9 of 26 , May 30, 2016


                      [APH]:

                      2. P = I

                      I1,I2,I3 = the Incenters of IBC,ICA,IAB resp.
                      I12,I13 = the reflections of I1 in AC,AB, resp.
                      I23,I21 = the reflections of I2 in BA,BC, resp.
                      I31,I32 = the reflections of I3 in CB,CA, resp.

                      The perpendicular bisectors of I12I13,I23I21,I31I32 are concurrent at ??? (on
                      the OI line)


                      Dear Angel and Cesar,

                      Thank you for your responses.

                      Now, I think the same is true if we replace the Incenters of the component triangles
                      with excenters. That is:

                      Denote:

                      J1 = the BC-excenter of IBC
                      J2 = the CA-excenter of ICA
                      J3 = the AB-excenter of IAB

                      J12,J13 = the reflections of J1 in AC,AB, resp.
                      J23,J21 = the reflections of J2 in BA,BC, resp.
                      J31,J32 = the reflections of J3 in CB,CA, resp.

                      The perpendicular bisectors of J12J13,J23IJ21,J31J32 are concurrent.

                      Antreas




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