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24545Re: Euler lines concurrency (X(1113) and X(1114))

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  • Antreas Hatzipolakis
    Oct 5, 2016

      Dear Seiichi Kirikami and Antreas Hatzipolakis,

      **** The Euler lines of AHX(1113), BHX(1113), CHX(1113) concur in U =((b^2+c^2-a^2) (R F1 + a^2 G1 |OH|):...:..),
              The Euler lines of AHX(1114), BHX(1114), CHX(1114) concur in V =( (b^2+c^2-a^2) (R F1 - a^2 G1 |OH|):...:..),

      where F1 = a^30 (b^2+c^2)-6 a^28 (2 b^4+b^2 c^2+2 c^4)+9 a^26 (6 b^6+5 b^4 c^2+5 b^2 c^4+6 c^6)-a^24 (111 b^8+245 b^6 c^2+42 b^4 c^4+245 b^2 c^6+111 c^8)+3 a^22 (21 b^10+242 b^8 c^2+41 b^6 c^4+41 b^4 c^6+242 b^2 c^8+21 c^10)+a^20 (174 b^12-1145 b^10 c^2-856 b^8 c^4+750 b^6 c^6-856 b^4 c^8-1145 b^2 c^10+174 c^12)-8 a^18 (47 b^14-81 b^12 c^2-306 b^10 c^4+150 b^8 c^6+150 b^6 c^8-306 b^4 c^10-81 b^2 c^12+47 c^14)+a^16 (207 b^16+982 b^14 c^2-4004 b^12 c^4+410 b^10 c^6+2794 b^8 c^8+410 b^6 c^10-4004 b^4 c^12+982 b^2 c^14+207 c^16)+a^14 (207 b^18-2387 b^16 c^2+4142 b^14 c^4+498 b^12 c^6-2076 b^10 c^8-2076 b^8 c^10+498 b^6 c^12+4142 b^4 c^14-2387 b^2 c^16+207 c^18)-4 a^12 (94 b^20-498 b^18 c^2+441 b^16 c^4+582 b^14 c^6-615 b^12 c^8+24 b^10 c^10-615 b^8 c^12+582 b^6 c^14+441 b^4 c^16-498 b^2 c^18+94 c^20)+a^10 (b^2-c^2)^4 (174 b^14+493 b^12 c^2-1435 b^10 c^4-1232 b^8 c^6-1232 b^6 c^8-1435 b^4 c^10+493 b^2 c^12+174 c^14)+a^8 (b^2-c^2)^4 (63 b^16-845 b^14 c^2+1176 b^12 c^4-123 b^10 c^6+322 b^8 c^8-123 b^6 c^10+1176 b^4 c^12-845 b^2 c^14+63 c^16)-a^6 (b^2-c^2)^6 (111 b^14-402 b^12 c^2-36 b^10 c^4+7 b^8 c^6+7 b^6 c^8-36 b^4 c^10-402 b^2 c^12+111 c^14)+a^4 (b^2-c^2)^8 (b^2+c^2)^2 (54 b^8-149 b^6 c^2+124 b^4 c^4-149 b^2 c^6+54 c^8)-6 a^2 (b^2-c^2)^10 (b^2+c^2)^3 (2 b^4-3 b^2 c^2+2 c^4)+(b^2-c^2)^12 (b^2+c^2)^4,

      and G1 = a^26 (b^4+c^4)-7 a^24 (b^6+b^4 c^2+b^2 c^4+c^6)+a^22 (18 b^8+59 b^6 c^2+16 b^4 c^4+59 b^2 c^6+18 c^8)-a^20 (14 b^10+205 b^8 c^2+73 b^6 c^4+73 b^4 c^6+205 b^2 c^8+14 c^10)+a^18 (-25 b^12+349 b^10 c^2+363 b^8 c^4-134 b^6 c^6+363 b^4 c^8+349 b^2 c^10-25 c^12)+a^16 (63 b^14-194 b^12 c^2-992 b^10 c^4+291 b^8 c^6+291 b^6 c^8-992 b^4 c^10-194 b^2 c^12+63 c^14)-2 a^14 (18 b^16+169 b^14 c^2-808 b^12 c^4+31 b^10 c^6+492 b^8 c^8+31 b^6 c^10-808 b^4 c^12+169 b^2 c^14+18 c^16)-2 a^12 (18 b^18-371 b^16 c^2+765 b^14 c^4+191 b^12 c^6-443 b^10 c^8-443 b^8 c^10+191 b^6 c^12+765 b^4 c^14-371 b^2 c^16+18 c^18)+a^10 (63 b^20-518 b^18 c^2+319 b^16 c^4+1488 b^14 c^6-1406 b^12 c^8+236 b^10 c^10-1406 b^8 c^12+1488 b^6 c^14+319 b^4 c^16-518 b^2 c^18+63 c^20)-a^8 (b^2-c^2)^2 (25 b^18+109 b^16 c^2-1104 b^14 c^4+1160 b^12 c^6-126 b^10 c^8-126 b^8 c^10+1160 b^6 c^12-1104 b^4 c^14+109 b^2 c^16+25 c^18)-a^6 (b^2-c^2)^4 (14 b^16-303 b^14 c^2+608 b^12 c^4-57 b^10 c^6+148 b^8 c^8-57 b^6 c^10+608 b^4 c^12-303 b^2 c^14+14 c^16)+a^4 (b^2-c^2)^6 (18 b^14-157 b^12 c^2+39 b^10 c^4+52 b^8 c^6+52 b^6 c^8+39 b^4 c^10-157 b^2 c^12+18 c^14)-a^2 (b^2-c^2)^8 (b^2+c^2)^2 (7 b^8-47 b^6 c^2+38 b^4 c^4-47 b^2 c^6+7 c^8)+(b^2-c^2)^10 (b^2+c^2)^3 (b^4-5 b^2 c^2+c^4)

      U = X(3)X(2575) /\ X(5)X(523)  and  V = X(3)X(2574) /\ X(5)X(523)

      **** The locus of P such that the Euler lines of PHA, PHB, PHC are concurrent is a circum-sextic  passing through the vertices of the circum-orthic triangle   and the point X(1263).

      Equation of sextic and figure in:
       http://amontes.webs.ull.es/ otrashtm/HGT2016.htm#HG051016

      Best regards
      Angel Montesdeoca

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