[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
A", B", C" = the reflections of A', B', C' in AP, BP, CP, resp.
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
D = the Poncelet point of ABCP.
DA" intersects again (Na) at A*
DB" intersects again (Nb) at B*
DC" intersects again (Nc) at C*.
A*, B*, C* and D are concyclic.
Center of the circle?
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[Ercole Suppa]
If P(u:v:w) (barys) the center W(P) of circle through A*, B*, C*, D is
W(P)= (b^2-c^2)^2 u^2 (v+w)^2 (b^8 (u+v)^2 w^2+c^8 v^2 (u+w)^2-b^6 c^2 (u+v) w (2 u^2+4 v w+u (v+w))-b^2 c^6 v (u+w) (2 u^2+4 v w+u (v+w))+2 b^4 c^4 v w (3 u^2+3 v w+2 u (v+w)))-a^10 v (u+v) w (u+w) (b^2 w (u^2 v+v (v-w) w+u (v^2+v w-w^2))+c^2 v (u^2 w+v w (-v+w)+u (-v^2+v w+w^2)))-a^6 (b^6 (u+v)^2 w^2 (v w^2 (v+w)+5 u^2 v (2 v+w)+u w (9 v^2+3 v w-w^2))+c^6 v^2 (u+w)^2 (v^2 w (v+w)+5 u^2 w (v+2 w)+u v (-v^2+3 v w+9 w^2))-b^2 c^4 v w (u^5 (4 v+5 w)+u v w^2 (-5 v^2+7 v w-3 w^2)+v^2 w^2 (2 v^2+v w-w^2)+u^4 (v^2+3 v w+3 w^2)+u^3 (5 v^3-12 v^2 w-v w^2+w^3)+u^2 w (-4 v^3-5 v^2 w-3 v w^2+3 w^3))-b^4 c^2 v w (u^5 (5 v+4 w)+u v^2 w (-3 v^2+7 v w-5 w^2)+u^4 (3 v^2+3 v w+w^2)+v^2 w^2 (-v^2+v w+2 w^2)+u^2 v (3 v^3-3 v^2 w-5 v w^2-4 w^3)+u^3 (v^3-v^2 w-12 v w^2+5 w^3)))-a^4 u (-b^8 (u+v)^2 w^2 (5 v w (v+w)+u (10 v^2+9 v w+w^2)) -c^8 v^2 (u+w)^2 (5 v w (v+w)+u (v^2+9 v w+10 w^2))+b^6 c^2 w (9 u^4 v (v+w)+u^3 (9 v^3+17 v^2 w+9 v w^2-3 w^3)+u^2 v (v^3+25 v^2 w+15 v w^2-w^3)-4 v^2 w (v^3-2 v^2 w-2 v w^2+w^3)+u v (v^4+13 v^3 w+14 v^2 w^2+7 v w^3-3 w^4))+b^2 c^6 v (9 u^4 w (v+w)-4 v w^2 (v^3-2 v^2 w-2 v w^2+w^3)+u^2 w (-v^3+15 v^2 w+25 v w^2+w^3)+u^3 (-3 v^3+9 v^2 w+17 v w^2+9 w^3)+u w (-3 v^4+7 v^3 w+14 v^2 w^2+13 v ^3+w^4))+b^4 c^4 v w (6 u^4 (v+w)-u^3 (v^2-10 v w+w^2)+2 v w (4 v^3-3 v^2 w-3 v w^2+4 w^3)+u^2 (7 v^3+3 v^2 w+3 v w^2+7 w^3)+u (2 v^4+v^3 w+10 v^2 w^2+v w^3+2 w^4)))+a^8 v w (b^4 (u+v)^2 w (u (7 v-w) w+2 v w^2+u^2 (5 v+w))+c^4 v (u+w)^2 (-u v (v-7 w)+2 v^2 w+u^2 (v+5 w))-b^2 c^2 (-2 u^4 v w+4 v^3 w^3+u^5 (v+w)+2 u v w (v^3+w^3)+2 u^3 (v^3-4 v^2 w-4 v w^2+w^3)+u^2 (3 v^4-3 v^3 w-8 v^2 w^2-3 v w^3+3 w^4)))-a^2 u (b^10 (u+v)^2 w^2 (v+w) (w (v+w)+u (5 v+2 w))+c^10 v^2 (u+w)^2 (v+w) (v (v+w)+u (2 v+5 w))-b^8 c^2 w (-v^2 (v-4 w) w (v+w)^2+u^3 (v+w)^2 (9 v+4 w)+u^4 (7 v^2+10 v w+2 w^2)+3 u v w (6 v^3+9 v^2 w+5 v w^2+2 w^3)+u^2 (2 v^4+31 v^3 w+40 v^2 w^2+13 v w^3+2 w^4))-b^2 c^8 v (v (4 v-w) w^2 (v+w)^2+u^3 (v+w)^2 (4 v+9 w) +u^4 (2 v^2+10 v w+7 w^2)+3 u v w (2 v^3+5 v^2 w+9 v w^2+6 w^3)+u^2 (2 v^4+13 v^3 w+40 v^2 w^2+31 v w^3+2 w^4))+b^6 c^4 w (-2 v^2 (2 v-3 w) w (v+w)^2+u^4 (2 v^2+3 v w-2 w^2)+u^3 (15 v^3+17 v^2 w+4 v w^2+2 w^3)+u^2 (3 v^4+22 v^3 w+24 v^2 w^2+6 v w^3+w^4)+2 u v (-v^4+9 v^3 w+9 v^2 w^2+2 v w^3+3 w^4)) +b^4 c^6 v (2 v (3 v-2 w) w^2 (v+w)^2+u^4 (-2 v^2+3 v w+2 w^2)+u^3 (2 v^3+4 v^2 w+17 v w^2+15 w^3)+2 u w (3 v^4+2 v^3 w+9 v^2 w^2+9 v w^3-w^4)+u^2 (v^4+6 v^3 w+24 v^2 w^2+22 v w^3+3 w^4))) : : (barys)
Pairs (P=X(i),W(P)=X(j)) for these {i,j}: {1,5},{524,6092},{1499, 6092},{2574, 113},{2575, 113},{3413,114},{3414,114} - analyzed up to point X(23110)
Some Pairs (P,W(P)):
P1= W(X(1)) = X(5) = N
P2 = W(X(2) = -8 a^10 (b^2+c^2)+4 a^8 (14 b^4+3 b^2 c^2+14 c^4)+a^6 (-112 b^6+b^4 c^2+b^2 c^4-112 c^6)+16 (b^2-c^2)^2 (b^8-4 b^6 c^2+5 b^4 c^4-4 b^2 c^6+c^8) +10 a^4 (12 b^8-13 b^6 c^2-6 b^4 c^4-13 b^2 c^6+12 c^8)-a^2 (72 b^10-237 b^8 c^2+149 b^6 c^4+149 b^4 c^6-237 b^2 c^8+72 c^10) : : (barys)
= (6-9-13) serch numbers [4.01586693977158804, 2.41828172794572675, 0.112992390358143026]
P3 = W(X(3)) =
= X(3)X(12278)∩X(5)X(1986)
Barycentrics a^2-b^2-c^2) ((b^2-c^2)^6 (b^2+c^2)+a^10 (b^4+c^4)-3 a^2 (b^2-c^2)^4 (b^4+b^2 c^2+c^4)-3 a^8 (b^6+c^6)+a^4 (b^2-c^2)^2 (2 b^6+b^4 c^2+b^2 c^4+2 c^6) +a^6 (2 b^8-b^6 c^2-b^2 c^6+2 c^8)) : :
= lies on these lines: {3,12278},{5,1986},{54,15059},{68,6640},{125,12038},{1352,6639},{2072,5449},{4549,18404},{5889,10255},{8548,11898},{10024,12162},{10254,15058},{10721,15062},{11585,12606}
= (6-9-13) serch numbers [2.42961930515895802, 2.78672848843828047, 0.590028156761435077]
P4 = W(X(4)) is indeterminate since W(X(4))= (0:0:0)
P5 = W(X(5)) = 2 a^24 (b^4+c^4)-19 a^22 (b^6+b^4 c^2+b^2 c^4+c^6)+(b^2-c^2)^10 (b^8-3 b^6 c^2+2 b^4 c^4-3 b^2 c^6+c^8)+a^20 (83 b^8+130 b^6 c^2+138 b^4 c^4+130 b^2 c^6+83 c^8) -a^2 (b^2-c^2)^8 (11 b^10-26 b^8 c^2+5 b^6 c^4+5 b^4 c^6-26 b^2 c^8+11 c^10)-a^18 (223 b^10+372 b^8 c^2+427 b^6 c^4+427 b^4 c^6+372 b^2 c^8+223 c^10)+a^4 (b^2-c^2)^6 (55 b^12 -110 b^10 c^2-14 b^8 c^4+53 b^6 c^6-14 b^4 c^8-110 b^2 c^10+55 c^12)+a^16 (417 b^12+537 b^10 c^2+563 b^8 c^4+562 b^6 c^6+563 b^4 c^8+537 b^2 c^10+417 c^12) -a^6 (b^2-c^2)^4 (167 b^14-315 b^12 c^2-54 b^10 c^4+103 b^8 c^6+103 b^6 c^8-54 b^4 c^10-315 b^2 c^12+167 c^14)-a^14 (582 b^14+260 b^12 c^2+183 b^10 c^4+145 b^8 c^6+145 b^6 c^8 +183 b^4 c^10+260 b^2 c^12+582 c^14)+a^8 (b^2-c^2)^2 (348 b^16-692 b^14 c^2+40 b^12 c^4+119 b^10 c^6+136 b^8 c^8+119 b^6 c^10+40 b^4 c^12-692 b^2 c^14+348 c^16) +a^12 (630 b^16-490 b^14 c^2-135 b^12 c^4-112 b^10 c^6-110 b^8 c^8-112 b^6 c^10-135 b^4 c^12-490 b^2 c^14+630 c^16)-a^10 (534 b^18-1218 b^16 c^2+530 b^14 c^4+63 b^12 c^6 +37 b^10 c^8+37 b^8 c^10+63 b^6 c^12+530 b^4 c^14-1218 b^2 c^16+534 c^18) : :2 a^24 (b^4+c^4)-19 a^22 (b^6+b^4 c^2+b^2 c^4+c^6)+(b^2-c^2)^10 (b^8-3 b^6 c^2+2 b^4 c^4-3 b^2 c^6+c^8)+a^20 (83 b^8+130 b^6 c^2+138 b^4 c^4+130 b^2 c^6+83 c^8) -a^2 (b^2-c^2)^8 (11 b^10-26 b^8 c^2+5 b^6 c^4+5 b^4 c^6-26 b^2 c^8+11 c^10)-a^18 (223 b^10+372 b^8 c^2+427 b^6 c^4+427 b^4 c^6+372 b^2 c^8+223 c^10)+a^4 (b^2-c^2)^6 (55 b^12 -110 b^10 c^2-14 b^8 c^4+53 b^6 c^6-14 b^4 c^8-110 b^2 c^10+55 c^12)+a^16 (417 b^12+537 b^10 c^2+563 b^8 c^4+562 b^6 c^6+563 b^4 c^8+537 b^2 c^10+417 c^12) -a^6 (b^2-c^2)^4 (167 b^14-315 b^12 c^2-54 b^10 c^4+103 b^8 c^6+103 b^6 c^8-54 b^4 c^10-315 b^2 c^12+167 c^14)-a^14 (582 b^14+260 b^12 c^2+183 b^10 c^4+145 b^8 c^6+145 b^6 c^8 +183 b^4 c^10+260 b^2 c^12+582 c^14)+a^8 (b^2-c^2)^2 (348 b^16-692 b^14 c^2+40 b^12 c^4+119 b^10 c^6+136 b^8 c^8+119 b^6 c^10+40 b^4 c^12-692 b^2 c^14+348 c^16) +a^12 (630 b^16-490 b^14 c^2-135 b^12 c^4-112 b^10 c^6-110 b^8 c^8-112 b^6 c^10-135 b^4 c^12-490 b^2 c^14+630 c^16)-a^10 (534 b^18-1218 b^16 c^2+530 b^14 c^4+63 b^12 c^6 +37 b^10 c^8+37 b^8 c^10+63 b^6 c^12+530 b^4 c^14-1218 b^2 c^16+534 c^18) : : (barys)
= (6-9-13) serch numbers [-1.01557077998224045, -2.22852337988860586, 5.65221333566829414]
P6 = W(X(6) = -2 a^14 (b^2+c^2)+(b^2-c^2)^4 (b^2+c^2)^2 (b^4-4 b^2 c^2+c^4)+a^12 (9 b^4+8 b^2 c^2+9 c^4)-a^10 (8 b^6+3 b^4 c^2+3 b^2 c^4+8 c^6) +a^8 (-9 b^8+10 b^6 c^2-52 b^4 c^4+10 b^2 c^6-9 c^8)-a^2 (b^2-c^2)^2 (4 b^10-11 b^8 c^2-7 b^6 c^4-7 b^4 c^6-11 b^2 c^8+4 c^10) +2 a^6 (7 b^10-17 b^8 c^2-10 b^6 c^4-10 b^4 c^6-17 b^2 c^8+7 c^10)-a^4 (b^12-8 b^10 c^2-13 b^8 c^4+72 b^6 c^6-13 b^4 c^8-8 b^2 c^10+c^12) : :(barys)
= (6-9-13) serch numbers [1.38382409134719299, 1.24977527675825046, 2.13674740199150125]
P7 = W(X(7)) = -6 a^6 (b+c)-a^5 (b^2+16 b c+c^2)-a^4 (b^3-37 b^2 c-37 b c^2+c^3)+5 (b-c)^4 (3 b^3+13 b^2 c+13 b c^2+3 c^3)-4 a^2 (b-c)^2 (14 b^3+47 b^2 c+47 b c^2+14 c^3) -a (b-c)^2 (5 b^4-34 b^3 c-22 b^2 c^2-34 b c^3+5 c^4)+6 a^3 (9 b^4+2 b^3 c-2 b^2 c^2+2 b c^3+9 c^4) : : (barys)
= (6-9-13) serch numbers [1.79090673548808780, 1.31173772447854221, 1.90596602550473244]
P8 = W(X(8)) = 6 a^9 (b+c)-a^8 (11 b^2+76 b c+11 c^2)+(b^2-c^2)^4 (15 b^2-46 b c+15 c^2)+18 a^7 (b^3+8 b^2 c+8 b c^2+c^3)+2 a^6 (9 b^4-43 b^3 c-212 b^2 c^2-43 b c^3 +9 c^4)-2 a^2 (b^2-c^2)^2 (17 b^4+81 b^3 c-284 b^2 c^2+81 b c^3+17 c^4)-2 a (b-c)^2 (b+c)^3 (18 b^4-125 b^3 c+222 b^2 c^2-125 b c^3+18 c^4)-2 a^5 (45 b^5+38 b^4 c -219 b^3 c^2-219 b^2 c^3+38 b c^4+45 c^5)+2 a^4 (6 b^6+185 b^5 c-78 b^4 c^2-370 b^3 c^3-78 b^2 c^4+185 b c^5+6 c^6)+2 a^3 (51 b^7-144 b^6 c-230 b^5 c^2+339 b^4 c^3 +339 b^3 c^4-230 b^2 c^5-144 b c^6+51 c^7) : : (barys)
= (6-9-13) serch numbers [3.37065923156406120, -0.364383995644243308, 2.33724144893928853]
Best regards
Ercole Suppa
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