[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
D = the Poncelet pouint of ABCP
DNa, DNb, DNc intersect the pedal circle of P ( = circumcircle of A'B'C') again at A*, B*, C*, resp.
The reflections of A'A*, B'B*, C'C* in AP, BP, CP, resp. are parallels.
Point of concurrence (on the line at infinuty) in terms of P?
---------------------------------------
[Ercole Suppa]:
Let P(u:v:w) be the barycentric coordinates of P.
Call La(P), Lb(P), Lc(P) the reflections of A'A*, B'B*, C'C* in AP, BP, CP, resp.
La(P), Lb(P), Lc(P) concur at the infinite point:
Q = c^4 u^3 v^2-a^2 b^2 u^3 v w+b^4 u^3 v w-a^2 c^2 u^3 v w+c^4 u^3 v w-a^4 u^2 v^2 w+b^4 u^2 v^2 w+a^2 c^2 u^2 v^2 w-2 b^2 c^2 u^2 v^2 w+c^4 u^2 v^2 w-a^4 u v^3 w+a^2 b^2 u v^3 w+b^4 u^3 w^2-a^4 u^2 v w^2+a^2 b^2 u^2 v w^2+b^4 u^2 v w^2-2 b^2 c^2 u^2 v w^2+c^4 u^2 v w^2-2 a^4 u v^2 w^2+a^2 b^2 u v^2 w^2+a^2 c^2 u v^2 w^2-a^4 v^3 w^2-a^4 u v w^3+a^2 c^2 u v w^3-a^4 v^2 w^3 : : (barys)
*** Some lines:
-- If P = I the lines La(P), Lb(P), Lc(P) are parallel to OI line
-- If P = G the lines La(P), Lb(P), Lc(P) are parallel to Brocard axis of the 3rd pedal triangle of X(6)
-- If P = O the lines La(P), Lb(P), Lc(P) are parallel to OH = Euler line
-- If P = N ( = NPC center of ABC) the lines La(P), Lb(P), Lc(P) are parallel to line X(3)X(14140).
They concur on the infinity line at the point
Q(P) = X(3)X(14140) ∩ X(5)X(252)
= 2 a^16-11 a^14 b^2+25 a^12 b^4-29 a^10 b^6+15 a^8 b^8+3 a^6 b^10-9 a^4 b^12+5 a^2 b^14-b^16-11 a^14 c^2+34 a^12 b^2 c^2-33 a^10 b^4 c^2+4 a^8 b^6 c^2+5 a^6 b^8 c^2+12 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2+25 a^12 c^4-33 a^10 b^2 c^4+4 a^8 b^4 c^4+a^6 b^6 c^4-2 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4-29 a^10 c^6+4 a^8 b^2 c^6+a^6 b^4 c^6-2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6+15 a^8 c^8+5 a^6 b^2 c^8-2 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+3 a^6 c^10+12 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-9 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : : (barys)
= 2 S^4 + (R^4+10 R^2 SB+10 R^2 SC-6 SB SC-6 R^2 SW-4 SB SW-4 SC SW+2 SW^2)S^2 -3 R^4 SB SC-2 R^2 SB SC SW+2 SB SC SW^2 : : (barys)
= {X[3]-X[14140], X[5]-X[252], X[12]-X[14102], X[30]-X[511], X[137]-X[24147], X[140]-X[6150], X[546]-X[16337], X[550]-X[14141], X[930]-X[6345], X[1263]-X[19553], X[3628]-X[10615], X[5501]-X[20414], X[10096]-X[10227], X[10126]-X[13856], X[10205]-X[14095], X[13372]-X[27090], X[22051]-X[27246], X[25043]-X[28237], X[27868]-X[30484]
= lies on these lines: {3,14140}, {5,252}, {12,14102}, {30,511}, {137,24147}, {140,6150}, {546,16337}, {550,14141}, {930,6345}, {1263,19553}, {3628,10615}, {5501,20414}, {10096,10227}, {10126,13856}, {10205,14095}, {13372,27090}, {22051,27246}, {25043,28237}, {27868,30484}
= isogonal conjugate of Q*(P)
= (6-9-13) search numbers: [0.97528232582918547049, 1.68798755678933419222, -1.61873707431377850407]
Q(P)* = ISOGONAL CONJUGATE OF Q(P)
= a^2 (a^16-6 a^14 b^2+16 a^12 b^4-26 a^10 b^6+30 a^8 b^8-26 a^6 b^10+16 a^4 b^12-6 a^2 b^14+b^16-5 a^14 c^2+17 a^12 b^2 c^2-21 a^10 b^4 c^2+9 a^8 b^6 c^2+9 a^6 b^8 c^2-21 a^4 b^10 c^2+17 a^2 b^12 c^2-5 b^14 c^2+9 a^12 c^4-12 a^10 b^2 c^4+2 a^8 b^4 c^4+2 a^6 b^6 c^4+2 a^4 b^8 c^4-12 a^2 b^10 c^4+9 b^12 c^4-3 a^10 c^6-5 a^8 b^2 c^6-a^6 b^4 c^6-a^4 b^6 c^6-5 a^2 b^8 c^6-3 b^10 c^6-15 a^8 c^8-4 a^6 b^2 c^8-4 a^4 b^4 c^8-4 a^2 b^6 c^8-15 b^8 c^8+29 a^6 c^10+33 a^4 b^2 c^10+33 a^2 b^4 c^10+29 b^6 c^10-25 a^4 c^12-34 a^2 b^2 c^12-25 b^4 c^12+11 a^2 c^14+11 b^2 c^14-2 c^16) (a^16-5 a^14 b^2+9 a^12 b^4-3 a^10 b^6-15 a^8 b^8+29 a^6 b^10-25 a^4 b^12+11 a^2 b^14-2 b^16-6 a^14 c^2+17 a^12 b^2 c^2-12 a^10 b^4 c^2-5 a^8 b^6 c^2-4 a^6 b^8 c^2+33 a^4 b^10 c^2-34 a^2 b^12 c^2+11 b^14 c^2+16 a^12 c^4-21 a^10 b^2 c^4+2 a^8 b^4 c^4-a^6 b^6 c^4-4 a^4 b^8 c^4+33 a^2 b^10 c^4-25 b^12 c^4-26 a^10 c^6+9 a^8 b^2 c^6+2 a^6 b^4 c^6-a^4 b^6 c^6-4 a^2 b^8 c^6+29 b^10 c^6+30 a^8 c^8+9 a^6 b^2 c^8+2 a^4 b^4 c^8-5 a^2 b^6 c^8-15 b^8 c^8-26 a^6 c^10-21 a^4 b^2 c^10-12 a^2 b^4 c^10-3 b^6 c^10+16 a^4 c^12+17 a^2 b^2 c^12+9 b^4 c^12-6 a^2 c^14-5 b^2 c^14+c^16) : : (barys)
= (68 R^2+4 SB+4 SC-20 SW)S^6 + (-142 R^6-116 R^4 SB-116 R^4 SC-92 R^2 SB SC+296 R^4 SW+64 R^2 SB SW+64 R^2 SC SW+20 SB SC SW-160 R^2 SW^2-8 SB SW^2-8 SC SW^2+24 SW^3)S^4 +(12 R^10-59 R^8 SB-59 R^8 SC-82 R^6 SB SC+53 R^8 SW-8 R^6 SB SW-8 R^6 SC SW-72 R^4 SB SC SW-14 R^6 SW^2+68 R^4 SB SW^2+68 R^4 SC SW^2+112 R^2 SB SC SW^2-48 R^4 SW^3-32 R^2 SB SW^3-32 R^2 SC SW^3-24 SB SC SW^3+28 R^2 SW^4+4 SB SW^4+4 SC SW^4-4 SW^5)S^2 -18 R^10 SB SC -21 R^8 SB SC SW+30 R^6 SB SC SW^2+16 R^4 SB SC SW^3-20 R^2 SB SC SW^4+4 SB SC SW^5 : : (barys)
= lies on the circumcircle and these lines: {110,15345}, {930,10205}, {1510,6345}
= a^2 (a^16-6 a^14 b^2+16 a^12 b^4-26 a^10 b^6+30 a^8 b^8-26 a^6 b^10+16 a^4 b^12-6 a^2 b^14+b^16-5 a^14 c^2+17 a^12 b^2 c^2-21 a^10 b^4 c^2+9 a^8 b^6 c^2+9 a^6 b^8 c^2-21 a^4 b^10 c^2+17 a^2 b^12 c^2-5 b^14 c^2+9 a^12 c^4-12 a^10 b^2 c^4+2 a^8 b^4 c^4+2 a^6 b^6 c^4+2 a^4 b^8 c^4-12 a^2 b^10 c^4+9 b^12 c^4-3 a^10 c^6-5 a^8 b^2 c^6-a^6 b^4 c^6-a^4 b^6 c^6-5 a^2 b^8 c^6-3 b^10 c^6-15 a^8 c^8-4 a^6 b^2 c^8-4 a^4 b^4 c^8-4 a^2 b^6 c^8-15 b^8 c^8+29 a^6 c^10+33 a^4 b^2 c^10+33 a^2 b^4 c^10+29 b^6 c^10-25 a^4 c^12-34 a^2 b^2 c^12-25 b^4 c^12+11 a^2 c^14+11 b^2 c^14-2 c^16) (a^16-5 a^14 b^2+9 a^12 b^4-3 a^10 b^6-15 a^8 b^8+29 a^6 b^10-25 a^4 b^12+11 a^2 b^14-2 b^16-6 a^14 c^2+17 a^12 b^2 c^2-12 a^10 b^4 c^2-5 a^8 b^6 c^2-4 a^6 b^8 c^2+33 a^4 b^10 c^2-34 a^2 b^12 c^2+11 b^14 c^2+16 a^12 c^4-21 a^10 b^2 c^4+2 a^8 b^4 c^4-a^6 b^6 c^4-4 a^4 b^8 c^4+33 a^2 b^10 c^4-25 b^12 c^4-26 a^10 c^6+9 a^8 b^2 c^6+2 a^6 b^4 c^6-a^4 b^6 c^6-4 a^2 b^8 c^6+29 b^10 c^6+30 a^8 c^8+9 a^6 b^2 c^8+2 a^4 b^4 c^8-5 a^2 b^6 c^8-15 b^8 c^8-26 a^6 c^10-21 a^4 b^2 c^10-12 a^2 b^4 c^10-3 b^6 c^10+16 a^4 c^12+17 a^2 b^2 c^12+9 b^4 c^12-6 a^2 c^14-5 b^2 c^14+c^16) : : (barys)
= (68 R^2+4 SB+4 SC-20 SW)S^6 + (-142 R^6-116 R^4 SB-116 R^4 SC-92 R^2 SB SC+296 R^4 SW+64 R^2 SB SW+64 R^2 SC SW+20 SB SC SW-160 R^2 SW^2-8 SB SW^2-8 SC SW^2+24 SW^3)S^4 +(12 R^10-59 R^8 SB-59 R^8 SC-82 R^6 SB SC+53 R^8 SW-8 R^6 SB SW-8 R^6 SC SW-72 R^4 SB SC SW-14 R^6 SW^2+68 R^4 SB SW^2+68 R^4 SC SW^2+112 R^2 SB SC SW^2-48 R^4 SW^3-32 R^2 SB SW^3-32 R^2 SC SW^3-24 SB SC SW^3+28 R^2 SW^4+4 SB SW^4+4 SC SW^4-4 SW^5)S^2 -18 R^10 SB SC -21 R^8 SB SC SW+30 R^6 SB SC SW^2+16 R^4 SB SC SW^3-20 R^2 SB SC SW^4+4 SB SC SW^5 : : (barys)
= lies on the circumcircle and these lines: {110,15345}, {930,10205}, {1510,6345}
= isogonal conjugate of Q(P)
-- If P = K (symmedian point) the lines La(P), Lb(P), Lc(P) are parallel to line X(2)X(6324)
Best regards,
Ercole Suppa
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου