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220182 points made from the concurrency of GK lines

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  • Antreas Hatzipolakis
    Dec 1, 2013
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      ---------- Forwarded message ----------
      From: Seiichi Kirikami
      Date: Sat, Nov 23, 2013 at 5:47 PM
      Subject: [EGML] 2 points made from the concurrency of GK lines
      To: Anopolis@yahoogroups.com



      Given a triangle ABC and the point X(13), we denote by Ba and Bc the intersections of the parallel line through X(13) with AB and CA respectively. The other intersections Cb, Ab, Ac and Bc are defined cyclically. GK lines of X(13)AcAb, X(13)BaBc and X(13)CbCa concur in a point Q, where G and K mean the centroid and the symmedian point of ABC respectively.

      If we use X(14) instead of X(13), we have another concurrent point R.

      Barycentrics for X(13) is as follows.

      x=8a^16+(-23(b^2+c^2)+4S’)a^14-(16(b^4+c^4)-73b^2c^2+26(b^2+c^2)S’)a^12+(134(b^6+c^6)-84b^2c^2(b^2+c^2)+46(b^4+c^4+b^2c^2)S’)a^10-(185(b^8+c^8)-34b^2c^2(b^4+c^4)-123b^4c^4+2(b^6+c^6+30b^2c^2(b^2+c^2))S’)a^8+(77(b^10+c^10)+52b^2c^2(b^6+c^6)-91b^4c^4(b^2+c^2)-(72(b^8+c^8)-86b^2c^2(b^4+c^4)+6b^4c^4)S’)a^6+(38(b^12+c^12)-132b^2c^2(b^8+c^8)+159b^4c^4(b^4+c^4)-130b^6c^6+(74(b^10+c^10)-74b^2c^2(b^6+c^6)+6b^4c^4(b^2+c^2))S’)a^4-(b^2-c^2)^2(44(b^10+c^10)-27b^2c^2(b^6+c^6)-17b^4c^4(b^2+c^2)+(26(b^8+c^8)-4b^2c^2(b^4+c^4)+4b^4c^4)S’)a^2+(b^2-c^2)^4(11(b^8+c^8)+9b^2c^2(b^4+c^4)-40b^4c^4+(2(b^6+c^6)-20b^2c^2(b^2+c^2))S’).

      S’ means 2*sqrt(3)* area of the triangle.

      kx=0.0368160852136.. (non-ETC).

      Barycentrics for X(14): replace S’ in baricentrics for x(13)by –S’.

      kx=3.15628332574…(non-ETC).

       

      Best regards,

      Seiichi Kirikami


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