HYACINTHOS 25559

[Antreas P. Hatzipolakis]:
   
Let ABC be a triangle, P, Q two isogonal conjugate points and A'B'C' the pedal triangle of Q.

Denote:

P, Pa, Pb, Pc = same points of ABC, PBC, PCA, PAB, resp.

R1 = the radical axis of the circles (Pb, PbA'), (Pc, PcA')
R2 = the radical axis of the circles (Pc, PcB'), (Pa, PaB')
R3 = the radical axis of the circles (Pa, PaC'), (Pb, PbC')

If R1, R2, R3 are concurrent, then P lies on the Euler line of ABC (and Pa, Pb, Pc on the Euler lines of PBC, PCA, PAB, resp. being same to P).

[Angel Montesdeoca]:

 

**** R1, R2, R3 concur always.
         If DEF is the cevian triangle of P(u:v:w),  PPa:PaD=v+w:u,  Pa=(u^2,v (2 u+v+w),w (2 u+v+w)) and R1, R2, R3 are concurrent at W=( (b^2-c^2) u (v-w)-a^2 (4 v w+u (v+w)): ... : ...).
         Pairs {P,W}:  {1, 354}, {2, 6}, {3, 51}, {4, 2}, {6, 373}, {13, 396}, {14, 395}, {20, 1853}, {30, 125}, {54, 5943}, {57, 374}, {58, 375}, {64, 3917}, {74, 511}, {76, 5306}, {83, 9300}, {84, 210}, {98, 524}, {99, 523}, {100, 9001}, {101, 9000}, {102, 8679}, {103, 674}, {104, 518}, {105, 9004}, {106, 9026}, {107, 9007}, {108, 9051}, {109, 8999}, {110, 8675}, {111, 9027}, {112, 520}, {254, 3167}, {262, 597}, {376, 4}, {378, 9730}, {393, 6090}, {476, 9003}, {477, 542}, {511, 6784}, {512, 6786}, {519, 3756}, {523, 5642}, {524, 6791}, {525, 6793}, {598, 3815}, {648, 9209}, {671, 230}, {691, 526}, {841, 5663}, {842, 2854}, {892, 9189}, {903, 3011}, {917, 9028}, {935, 9033}, {953, 2810}, {1105, 11245}, {1113, 2574}, {1114, 2575}, {1138, 110}, {1141, 5965}, {1173, 6688}, {1217, 11402}, {1289, 8057}, {1292, 513}, {1293, 9002}, {1294, 1503}, {1295, 3827}, {1296, 512}, {1297, 2393}, {1300, 3564}, {1327, 590}, {1328, 615}, {1389, 3742}, {1494, 468}, {1989, 3292}, {2396, 6041}, {2687, 2836}, {2691, 8674}, {2693, 2781}, {2696, 690}, {2709, 9023}, {2710, 2871}, {2723, 5848}, {2724, 5845}, {2737, 900}, {2966, 1637}, {3227, 3290}, {3228, 3291}, {3296, 3475}, {3346, 154}, {3424, 599}, {3426, 5650}, {3431, 5640}, {3563, 8681}, {3565, 924}, {4235, 1640}, {5485, 7735}, {5627, 5972}, {6011, 9013}, {6233, 9044}, {6236, 3906}, {7576, 1209}, {7607, 8584}, {7612, 1992}, {9302, 385}, {10098, 9517}, {10101, 2850}, {10308, 3740}, {11270, 3060}.
        
**** The locus of point of concurrence of R1, R2, R3 as P moves on the Euler line of ABC is the rectangular hyperbola that passes through X(2), X(4), X(6), X(51), X(125), X(1209), X(1640), X(1853), X(2574), X(2575), X(9730)  and the asymtotes are parallel  to the asymtots of  the Jerabek hyperbola.

This center is
 
(a^2 (-2 a^2 (b^2-c^2)^4-a^8 (b^2+c^2)-a^4 b^2 c^2 (b^2+c^2)+2 a^6 (b^4+c^4)+(b^2-c^2)^2 (b^6-4 b^4 c^2-4 b^2 c^4+c^6)): ... : ...),

the midpoint of  X(51) and X(125), with search numbers (1.80544864219486, 1.46178909979057, 1.79537265488558).

Angel Montesdeoca