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22486Fwd: [EGML] 2 circles and 5 lines in a right triangle [2 Attachments]

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  • Antreas Hatzipolakis
    Jul 2, 2014
    • 0 Attachment


      ---------- Forwarded message ----------
      From: Seiichi Kirikami
      Date: Wed, Jul 2, 2014 at 5:16 AM
      Subject: [EGML] 2 circles and 5 lines in a right triangle [2 Attachments]
      To: "anopolis@yahoogroups.com" <anopolis@yahoogroups.com>


       
      [Attachment(s) from skirikami_0623@... included below]

      Dear friends,
       
      [1] Given a triangle ABC right in A, its incircle(I), its circumcircle(O), and its excircles (Ia), (Ib), (Ic) with their centers I, O, Ia, Ib, Ic, denote:
      S, Q, R= the contact points of (I) with BC, CA, AB respectively,
      Aa, Ba, Ca=the contact points of (Ia) with BC, CA, AB respectively,
      Ab, Bb, Cb=the contact points of (Ib) with BC, CA, AB respectively,
      Ac, Bc, Cc=the contact points of (Ic) with BC, CA, AB respectively,
      B’, C’=the midpoints of CA, AB respectively,
      B’’, C’’=the intersections of B’C’ with (O) respectively.
      1. 8 points B’’, Q, R, C’’, Ac, Ca, Ba, and Ab are on a circle (1-circle).
      2. 8 points B’’, Bb, Aa, S, Cc, C’’, Bc, and Cb are on another circle (2-circle).
      See the attached pictures.
      This is an extension of Le Probleme de Toshio Seimiya in Quelques Theoremes Oublies, vol. 1 (2007), Geometrie by Jean-Louis Ayme.
       
      [2] Given a triangle ABC right in A, its excircles (Ia), (Ib), (Ic) with their centers Ia, Ib, Ic, and its angle bisectors (A-bsc), (B-bsc), (C-bsc), denote:
      B’, C’=the midpoints of CA, AB respectively,
      Ka, Kb, Kc=the intersections farthest from the sides of ABC of (A-bsc), (B-bsc), (C-bsc) with (Ia), (Ib), (Ic) respectibvely,
      Jb, Jc=the intersections nearest to the sides of ABC of (B-bsc), (C-bsc) with (Ib), (Ic),
      Mb, Mc=the intersections farthest from BC of (Ia) with the angle bisector of CBCa, BCBa respectively,
      Lb, Lc=the intersections nearest to BC of (Ia) with the angle bisector of CBCa, BCBa respectively.
      1. Kb, B’, Lb are on a line (1-line),
      2. Kc, C’, Lc are on a line (2-line),
      3. Mc, C’, Jc are on a line (3-line),
      4. Mb, B’, Jb are on a line (4-line),
      P= the intersection of 1-line and 2-line,
      Q=the intersection of 3-line and 4-line.
           5.P, Q, Ka are on a line (5-line).
      See the attached pictures.
      This is the continuation of Anopolis message #817.

      Best regards,
      Seiichi.