Deonte:
M1, M2, M3 = the midpoints of AP, BP, CP, resp.
The triangles:
The cyclologic center (MaMbMc, M1M2M3) is the Poncelet point of (ABCP)
[circumcircles of MaM2M3, MbM3M1, McM1M2 = NPCs of PBC, PCA, PAB, resp.]
Let U be the cyclologic center (M1M2M3, MaMbMc)
2. M1M2M3, PaPbPc are cyclologic.
The cyclologic center (PaPbPc, M1M2M3) is the Poncelet point of (ABCP)
Which is the line UW ?
**************
Denote:
Which is the locus of P such that OaObOc, O1O2O3 are perspective ?
The entire plane? (Perspector in terms of P?)
[César Lozada]:
> Which is the line UW ?
For P not in the infinity and not on the circumcircle, the line UV is the isogonal conjugate of the circumconic centered at Q, where:
Q = Complement(AntigonalConjugate( Complement( AnticomplementaryConjugate( AntigonalConjugate(P)))))
ETC pairs (P,Q(P)): (3,14993), (23,2), (69,15477), (76,9467), (265,11597), (316,206), (671,6593), (858,3162), (895,2482), (1320,214), (1325,10), (1916,8290), (2071,6523), (3484,14363), (5011,9), (5179,478), (5523,6), (10152,1511), (11607,513), (11610,5976), (13509,1249), (14366,523), (14887,514), (15342,1084), (15343,8054)
Some others:
Q( X(1) ) = midpoint of X(1168) and X(2222)
= a*(a^5-(b^2-b*c+c^2)*a^3+(b+c) *(b^2-3*b*c+c^2)*a^2-(b^2-4*b* c+c^2)*b*c*a-(b^3+c^3)*(b-c)^ 2)*(a^2-c*a-b^2+c^2)*(a^2-b*a+ b^2-c^2) : : (barys)
= on lines: {80, 3465}, {106, 1168}, {519, 1807}, {1324, 6187}
= midpoint of X(1168) and X(2222)
= [ -0.0566339426909219, 2.7453881035796360, 1.7661499222096710 ]
Q(X(2)) = complement of X(13574)
= (-SA*SW*(3*SA-2*SW)+(9*R^2-3* SW)*S^2)*(SB+SC)*(3*SB-SW)*(3* SC-SW) : : (barys)
= X(23)-3*X(11580)
= on the cubic K043 and lines: {2, 8877}, {23, 111}, {67, 524}, {468, 8753}, {575, 10558}, {671, 5189}, {892, 3266}, {897, 7292}, {1551, 14094}, {10559, 11422}
= midpoint of X(691) and X(10630)
= complement of X(13574)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (23, 15398, 111), (10630, 11580, 111)
= [ -0.2215188146043134, 3.6701379669870000, 1.2020391884261390 ]
Q(X(6)) ) = complement of X(14364)
= (3*a^8-2*(b^2+c^2)*a^6-(2*b^4- 3*b^2*c^2+2*c^4)*a^4+(b^2+c^2) *(2*b^4-3*b^2*c^2+2*c^4)*a^2-( b^4-c^4)^2)*(a^4-c^2*a^2+c^4- b^4)*(a^4-b^2*a^2+b^4-c^4) : : (barys)
= on the cubic K043 and lines: {2, 14364}, {6, 14357}, {67, 187}, {111, 468}, {524, 10317}, {3455, 5938}
= midpoint of X(935) and X(10415)
= complement of X(14364)
= [ 1.2084557160378820, 1.4118577966469470, 2.1054756768267010 ]
O1O2O3 are perspective ?> Which is the locus of P such that OaObOc,
> The entire plane? (Perspector in terms of P?)
Yes. The locus is the entire plane.
For P = u : v : w (trilinears), the perspector Z(P) is:
Z(P) = b*c*(-a^2+b^2+c^2)*(a^6-b^2*a^ 4+(b^2-c^2)*(b^2+2*c^2)*a^2-( b^2-c^2)^3)*u^2*v^2-b*c*(-a^2+ b^2+c^2)*(a^6-c^2*a^4-(b^2-c^ 2)*(2*b^2+c^2)*a^2+(b^2-c^2)^ 3)*u^2*w^2+(b^2-c^2)*b*c*a^2* v^2*w^2*(-a^2+b^2+c^2)^2-a*b* c^2*(-a^2+b^2+c^2)*((b^2+c^2)* a^2-(b^2-c^2)^2)*u*w^3+a*b^2* c*(-a^2+b^2+c^2)*((b^2+c^2)*a^ 2-(b^2-c^2)^2)*v^3*u-c^2*a^2*( -a^2+b^2+c^2)*(a^4-(b^2+2*c^2) *a^2-(b^2-c^2)*c^2)*v*w^3-c*a* ((b^2-c^2)*a^6-(2*b^2-3*c^2)*( b^2+c^2)*a^4+(b^2-c^2)*(b^4+2* b^2*c^2+3*c^4)*a^2-(b^2-c^2)^ 3*c^2)*u^3*v-a*b*((b^2-c^2)*a^ 6-(3*b^2-2*c^2)*(b^2+c^2)*a^4+ (b^2-c^2)*(3*b^4+2*b^2*c^2+c^ 4)*a^2-(b^2-c^2)^3*b^2)*u^3*w+ b^2*a^2*(-a^2+b^2+c^2)*(a^4-( 2*b^2+c^2)*a^2+(b^2-c^2)*b^2)* w*v^3+c*a*(2*a^8-5*(b^2+c^2)* a^6+(5*b^4+4*b^2*c^2+3*c^4)*a^ 4-(3*b^2-c^2)*(b^4+c^4)*a^2+( b^4+2*b^2*c^2-c^4)*(b^2-c^2)^ 2)*u*v*w^2-a*b*(2*a^8-5*(b^2+ c^2)*a^6+(3*b^4+4*b^2*c^2+5*c^ 4)*a^4+(b^2-3*c^2)*(b^4+c^4)* a^2-(b^4-2*b^2*c^2-c^4)*(b^2- c^2)^2)*u*v^2*w-(b^2-c^2)*v*w* u^2*(3*a^8-9*(b^2+c^2)*a^6+(9* b^4+8*b^2*c^2+9*c^4)*a^4-3*(b^ 2+c^2)*(b^4+c^4)*a^2+4*(b^2-c^ 2)^2*b^2*c^2) : : (trilinears)
Z(P) lies on the line {X(3), P}
· If P lies on the Euler line and not P in {O,H} then Z(P) = X(5) = N
· If P lies on the Feuerbach-Napoleon cubic K005 and not P in {O,H} then Z(P)=P (ETC centers on K005: 1, 3, 4, 5, 17, 18, 54, 61, 62, 195, 627, 628, 2120, 2121, 3336, 3459, 3460, 3461, 3462, 3463, 3467, 3468, 3469, 3470, 3471, 3489, 3490, 6191, 6192, 7344, 7345, 8837, 8839, 8918, 8919, 8929, 8930)
· If P lies on the Lemoine cubic (K009) and not P in {O,H} then Z(P) is the point at infinity of line {X(3), P}. (ETC centers on K009: 3, 4, 32, 56, 1147, 6177, 6178, 6337, 7367, 10570, 11517, 13608, 14357, 14376, 14377, 14378, 14379, 14381, 14382, 14383, 14384, 14385, 14386, 15261, 15454)
ETC pairs (P, Z(P)) for other cases:
(6,575), (8,5690), (13,17), (14,18), (15,61), (16,62), (36,65), (65,5885), (265,5449), (399,3470), (484,3336), (895,8548), (1138,3471), (1157,195), (1173,15047), (1263,3459), (3065,3467), (3465,3469), (3466,3468), (3483,3461), (3484,3463), (5903,65), (6145,14076), (7165,3460), (14483,5643)
Example:
Z(X(7)) = {3, 7}/\{5, 8255}
= 16*q*p^5 - 4*(6*q^2-5)*p^4 - 52*q*p^3 + (4*q^2-5)*(2*q^2+7)*p^2 + (14*q^2+31)*q*p + 14 +8*q^2 : : (trilinears), where p=sin(A/2), q=cos((B-C)/2)
= 2*(b+c)*a^8-5*(b^2+c^2)*a^7-
(b+c)*(b^2+14*b*c+c^2)*a^6 + (11*b^4+11*c^4+3*(3*b^2+2*b*c+ 3*c^2)*b*c)*a^5 –
(b+c)*(5*b^4+5*c^4-2*(13*b^2+ b*c+13*c^2)*b*c)*a^4-(7*b^4+7* c^4+2*(13*b^2+18*b*c+13*c^2)* b*c)*(b-c)^2*a^3 +
(b^2-c^2)*(b-c)*(5*b^4+5*c^4- 2*(3*b^2+7*b*c+3*c^2)*b*c)*a^2 +
(b^2-c^2)^2*(b-c)^2*(b^2+5*b* c+c^2)*a-(b^2-c^2)^3*(b-c)^3 : : (barys)
= r*(7*R+2*r)*X(3)+(8*R^2+7*R*r+ 2*r^2)*X(7)
= on lines: {3, 7}, {5, 8255}, {7701, 11374}
= [ 1.6443480812816350, 1.5542656130896950, 1.8057045584076810 ]
César Lozada
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