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Homothetic triangles

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  • Antreas
    Let ABC, A B C be two homothetic triangles and X,Y,Z three collinear points. Are the triangles bounded by the lines (AX, BY, CZ) and (A X, B Y, C Z)
    Message 1 of 8 , Apr 30, 2010
      Let ABC, A'B'C' be two homothetic triangles
      and X,Y,Z three collinear points.

      Are the triangles bounded by the lines
      (AX, BY, CZ) and (A'X, B'Y, C'Z) perspective?

      APH
    • Antreas
      ... As Francisco pointed out: this follows inmediately from Desargues theorem for any ABC and A B C We can use it in the reference triangle ABC to get
      Message 2 of 8 , May 1, 2010
        --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
        >
        > Let ABC, A'B'C' be two homothetic triangles
        > and X,Y,Z three collinear points.
        >
        > Are the triangles bounded by the lines
        > (AX, BY, CZ) and (A'X, B'Y, C'Z) perspective?

        As Francisco pointed out: "this follows inmediately from Desargues
        theorem for any ABC and A'B'C'"

        We can use it in the reference triangle ABC to get points,
        from two given triangles and three collinear points.

        Here is an example:

        Inside ABC exists a unique point P such that:
        Three Congruent (equal) circles (A'),(B'),(C') concur at P,
        and circle (A') touches the sides of the angle A,
        (B') of angle B and (C') of angle C.
        [A', B', C' are the centers of the circles]

        Now, let Lp a line passing through P and
        intersecting the circles (A'),(B'),(C') at
        X,Y,Z resp. (other than P).

        The triangles bounded by the lines
        (AX, BY, CZ) and (A'X, B'Y, C'Z) are perspective.

        Which is the locus of the perspectors
        as L moves around P?

        APH
      • Barry Wolk
        ... I get two such points P. If your 3 circles have radius r*R/(R+r) then P=X(55). And if their radius is r*R/(R-r) then P=X(56). -- Barry Wolk
        Message 3 of 8 , May 6, 2010
          Antreas wrote:
          > Here is an example:
          >
          > Inside ABC exists a unique point P such that:
          > Three Congruent (equal) circles (A'),(B'),(C') concur at P,
          > and circle (A') touches the sides of the angle A,
          > (B') of angle B and (C') of angle C.
          > [A', B', C' are the centers of the circles]

          I get two such points P. If your 3 circles have radius r*R/(R+r)
          then P=X(55). And if their radius is r*R/(R-r) then P=X(56).
          --
          Barry Wolk
        • Antreas Hatzipolakis
          ... inside triangle (that s what I had in mind) APH [Non-text portions of this message have been removed]
          Message 4 of 8 , May 6, 2010
            On Thu, May 6, 2010 at 11:18 PM, Barry Wolk <wolkbarry@...> wrote:

            >
            >
            > Antreas wrote:
            > > Here is an example:
            > >
            > > Inside ABC exists a unique point P such that:
            > > Three Congruent (equal) circles (A'),(B'),(C') concur at P,
            > > and circle (A') touches the sides of the angle A,
            > > (B') of angle B and (C') of angle C.
            > > [A', B', C' are the centers of the circles]
            >
            > I get two such points P. If your 3 circles have radius r*R/(R+r)
            > then P=X(55). And if their radius is r*R/(R-r) then P=X(56).
            > --
            > Barry Wolk
            >
            >
            > Indeed. Two points P. But I think only one point with circles centers
            inside triangle (that's what I had in mind)

            APH


            [Non-text portions of this message have been removed]
          • Antreas Hatzipolakis
            Let ABC be a triangle and A B C the cevian triangle of I. Denote: Bc, Cb = the reflections of B, C in CC , BB , resp. Oa = the circumcenter of ABcCb.
            Message 5 of 8 , Jul 25, 2015
              Let ABC be a triangle and A'B'C' the cevian triangle of I.

              Denote:

              Bc, Cb = the reflections of B, C in CC', BB', resp.

              Oa = the circumcenter of ABcCb. Similarly Ob, Oc.

              OaObOc and the excentral triangle IaIbIc are homothetic.

              Which point is the homothetic center wrt triangles:

              1. ABC (on its OI line)

              2. OaObOc (on its Euler line)

              3. IaIbIc (on its Euler line)  ??

              APH
            • Antreas Hatzipolakis
              [APH]: Let ABC be a triangle and A B C the cevian triangle of I. ... Locus: Let ABC be a triangle P a point and PaPbPc the antipedal triangle of P. Denote:
              Message 6 of 8 , Jul 25, 2015


                [APH]:

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

                Denote:

                Bc, Cb = the reflections of B, C in CC', BB', resp.

                Oa = the circumcenter of ABcCb. Similarly Ob, Oc.

                OaObOc and the excentral triangle IaIbIc are homothetic.

                Which point is the homothetic center wrt triangles:

                1. ABC (on its OI line)

                2. OaObOc (on its Euler line)

                3. IaIbIc (on its Euler line)  ??

                APH


                Locus:

                Let ABC be a triangle P a point and PaPbPc the antipedal triangle of P.

                Denote:

                Bc, Cb = the reflections of B, C in CP, BP, resp.

                Oa = the circumcenter of ABcCb. Similarly Ob,Oc.

                Which is the locus of P such that OaObOc, PaPbPc are perspective?

                APH




              • Antreas Hatzipolakis
                [APH]: Let ABC be a triangle and A B C the cevian triangle of I. Denote: Bc, Cb = the reflections of B, C in CC , BB , resp. Oa = the circumcenter of ABcCb.
                Message 7 of 8 , Jul 26, 2015

                  [APH]:

                   

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

                  Denote:

                  Bc, Cb = the reflections of B, C in CC', BB', resp.

                  Oa = the circumcenter of ABcCb. Similarly Ob, Oc.

                  OaObOc and the excentral triangle IaIbIc are homothetic.

                  Which point is the homothetic center wrt triangles:

                  1. ABC (on its OI line)

                  2. OaObOc (on its Euler line)

                  3. IaIbIc (on its Euler line)  ??

                  APH



                  [César Lozada]


                  1. ABC (on its OI line)
                  X(3576)

                  2. OaObOc (on its Euler line)
                  X(381)

                  3. IaIbIc (on its Euler line)   
                  X(381)

                  CL





                • Antreas Hatzipolakis
                  Lemma: Let x, y be two perpendicular lines and L a line. The reflections of L in x, y are parallels. Corollary: Let ABC be a triangle, x,y,z three lines
                  Message 8 of 8 , Nov 1, 2017
                    Lemma:

                    Let x, y be two perpendicular lines and L a line.
                    The reflections of L in x, y are parallels.

                    Corollary:

                    Let ABC be a triangle, x,y,z three lines perpendicular to BC, CA, AB, resp. and L1, L2, L3 three lines.

                    Denote:

                    La, Lx = the reflections of L1 in BC, x, resp.
                    Lb, Ly = the reflections of L2 in CA, y, resp.
                    Lc, Lz = the reflections of L3 in AB, z, resp.

                    AaBbCc = the triangle bounded by La, Lb, Lc, resp.
                    AxByCz = the triangle bounded by Lx, Ly, Lz, resp.

                    AaBbCc, AxByCz are homothetic.


                    Application:

                    Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

                    Denote:

                    L1, L2, L3 = the Euler lines of AB'C', BC'A', CA'B', resp.

                    La, Lb, Lc = the reflections of L1, L2, L3 in BC, CA, AB, resp.
                    Li, Lii, Liii = the reflections of L1, L2, L3 in PA', PB', PC', resp.

                    AaBbCc = the triangle bounded by La,Lb,Lc
                    AiBiiCiii = the triangle bounded by Li, Lii, Liii

                    Which is the locus of the homothetic center of AaBbCc, AiBiiCiii as P moves on a line, the Euler line for example?

                    APH
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