[Antreas P. Hatzipolakis]:
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
A', B', C' = the orthogonal projections of Na, Nb, Nc on BC, CA, AB, resp.
Ab, Ac = the orthogonal projections of Na on B'Nb, C'Nc, resp.
La, Lb, Lc = the Euler lines of NaAbAc, NbBcBa, NcNCaCb, resp,
A*B*C* = the triangle bounded by La, Lb, Lc
ABC, A*B*C* are parallelogic.
The parallelogic center (ABC, A*B*C*) is fixed point on the circumcircle.
The other one (A*B*C*, ABC) in terms of P?
PS. A question is: which is the locus of P such that La, Lb, Lc are concurrent?
[César Lozada]:
>> PS. A question is: which is the locus of P such that La, Lb, Lc are concurrent?
Locus = {Linf} ∪ {circumcircle} ∪ {Jerabek hyperbola}
>>The parallelogic center (ABC, A*B*C*) is fixed point on the circumcircle.
At X(74)
>>The other one (A*B*C*, ABC)=Q*(P) in terms of P?
For P=x:y:z (barys),
Q*(P) = x*((c^2*a^8-2*a^6*b^2*c^2+c^4*(4*b^2-3*c^2)*a^4+2*(b^2-c^2)*a^2*c^2*(-c^4-b^2*c^2+b^4)-(b^2-c^2)^2*b^2*c^2*(b^2+2*c^2))*x*y^2+(a^8*b^2-2*a^6*b^2*c^2-b^4*(3*b^2-4*c^2)*a^4+2*(b^2-c^2)*a^2*b^2*(-c^4+b^2*c^2+b^4)-(b^2-c^2)^2*b^2*c^2*(c^2+2*b^2))*x*z^2+(a^10+(-2*b^2-4*c^2)*a^8+(5*c^4+2*b^4+2*b^2*c^2)*a^6+(b^4*c^2-3*b^2*c^4-c^6-b^6)*a^4-(b^2-c^2)^2*a^2*(b^4-2*b^2*c^2+2*c^4)+(b^2+c^2)*(b^2-c^2)^4)*y*z^2+(a^10+(-4*b^2-2*c^2)*a^8+(5*b^4+2*b^2*c^2+2*c^4)*a^6+(-b^6+b^2*c^4-3*b^4*c^2-c^6)*a^4-(b^2-c^2)^2*a^2*(2*b^4-2*b^2*c^2+c^4)+(b^2+c^2)*(b^2-c^2)^4)*z*y^2+(2*a^10+(-7*b^2-7*c^2)*a^8+(8*b^2*c^2+7*c^4+7*b^4)*a^6+(b^2+c^2)*(c^2+3*b*c+b^2)*(c^2-3*b*c+b^2)*a^4-(b^2-c^2)^2*a^2*(5*b^4-4*b^2*c^2+5*c^4)+(b^2-c^2)^2*(b^2+c^2)*(2*c^4-7*b^2*c^2+2*b^4))*x*y*z) : :
ETC centers (P,Q*(P)): (3,140), (4,5), (6,3589), (54,6689), (64,6696), (65,3812), (66,6697), (67,6698), (68,5449), (69,141), (72,5044), (74,6699), (110,16534), (250,5972), (265,20304), (895,15118), (5466,14566), (5486,16511)
If P lies on Linf, then Q(P)=P
If P lies on the circumcircle, then Q*(P) is the midpoint of P and X(113)
If P lies on the Jerabek hyperbola, then PQ* = (3/4)*PG = complement-of-complement-of-P
Some others:
Q*( X(1) ) = name pending
= (-a+b+c)*(2*a^9+(b+c)*a^8-(5*b^2-6*b*c+5*c^2)*a^7-(b+c)*(3*b^2-4*b*c+3*c^2)*a^6+2*(b^4+c^4-(3*b^2-5*b*c+3*c^2)*b*c)*a^5+(b^3+c^3)*(2*b^2-3*b*c+2*c^2)*a^4+(3*b^4+3*c^4+(b^2-7*b*c+c^2)*b*c)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(b^4+c^4+(b^2-3*b*c+c^2)*b*c)*a^2-(b^2-c^2)^2*(2*b^4+2*c^4-(5*b^2-3*b*c+5*c^2)*b*c)*a-(b^4-c^4)*(b^2-c^2)^2*(b-c)) : : (barys)
= lies on the line {758, 13464}
= [ 3.3783807561137770, 1.0806052138382560, 1.3333005233515360 ]
Q*( X(2) ) = X(2)X(525) ∩ X(5)X(524)
= 2*a^10-6*(b^2+c^2)*a^8+(7*b^4+4*b^2*c^2+7*c^4)*a^6-2*(b^2+c^2)*(b^4+c^4)*a^4-(b^2-c^2)^2*(3*b^4-7*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4-7*b^2*c^2+2*c^4) : : (barys)
= (108*R^2+3*SA-25*SW)*S^2+6*R^2*(3*SA+SW)*(3*SA-2*SW)-3*SW*(4*SA^2-SA*SW-SW^2) : : (barys)
= lies on these lines: {2, 525}, {5, 524}
= [ 4.9607612182217720, 0.3917781511950590, 1.0798513534392900 ]
Q*( X(5) ) = X(1154)X(3850) ∩ X(3628)X(6368)
= ((51*R^2+2*SA-12*SW)*S^2+72*R^6-2*(36*SA+35*SW)*R^4+(18*SA^2+40*SA*SW+13*SW^2)*R^2-2*(2*SA+3*SW)*SA*SW)*(S^2+SB*SC) : : (barys)
= lies on these lines: {1154, 3850}, {3628, 6368}
= [ -2.7540818016187290, -1.2158685468644050, 5.7535343074068410 ]
Q*( X(98) ) = MIDPOINT OF X(98) AND X(113)
= (108*R^2+3*SA-23*SW)*S^4+(6*R^2*(9*SA^2-3*SA*SW-7*SW^2)-SW*(9*SA^2-10*SW^2))*S^2+(18*R^2-5*SW)*SB*SC*SW^2 : : (barys)
= 3*X(2)+X(22265), X(110)+3*X(14651), 3*X(125)-X(15545), X(148)+3*X(15035), X(399)+3*X(14849), 7*X(3526)-3*X(14850), X(11005)-5*X(14061), X(11005)-3*X(23515), 2*X(11623)+X(16534), 3*X(11656)-X(22265), X(12188)+3*X(14643), X(12295)-3*X(14639), X(13172)-5*X(15051), 5*X(14061)-3*X(23515), 5*X(15059)-X(18331), X(15545)+3*X(18332), 2*X(20397)+X(31854)
= lies on these lines: {2, 11656}, {3, 16278}, {5, 542}, {98, 113}, {110, 14651}, {114, 12900}, {115, 17702}, {125, 15545}, {148, 15035}, {399, 14849}, {541, 5465}, {690, 6036}, {2777, 12042}, {2782, 5972}, {3526, 14850}, {3906, 16760}, {5642, 11632}, {6321, 16163}, {11005, 14061}, {12188, 14643}, {12295, 14639}, {13172, 15051}, {15059, 18331}, {15303, 19905}, {15342, 16003}, {15357, 20397}
= midpoint of X(i) and X(j) for these {i,j}: {2, 11656}, {3, 16278}, {98, 113}, {125, 18332}, {5465, 6055}, {5642, 11632}, {6321, 16163}, {15303, 19905}, {15342, 16003}, {15357, 31854}
= reflection of X(i) in X(j) for these (i,j): (114, 12900), (6699, 6036), (15357, 20397), (15359, 20398)
= complement of the complement of X(22265)
= {X(11005), X(14061)}-harmonic conjugate of X(23515)
= [ 3.8707340581660790, 4.7953915826300260, -1.4657146406053680 ]
Q*( X(99) ) = MIDPOINT OF X(99) AND X(113)
= 2*a^14-10*(b^2+c^2)*a^12+(17*b^4+26*b^2*c^2+17*c^4)*a^10-10*(b^2+c^2)^3*a^8-(3*b^8+3*c^8-(19*b^4+18*b^2*c^2+19*c^4)*b^2*c^2)*a^6+(b^2+c^2)*(b^4+c^4)*(7*b^4-17*b^2*c^2+7*c^4)*a^4-(b^2-c^2)^2*(4*b^8+4*c^8+(b^2+c^2)^2*b^2*c^2)*a^2+(b^6-c^6)*(b^2-c^2)^2*(b^4-c^4) : : (barys)
= (36*R^2+3*SA-7*SW)*S^4-(6*R^2*(9*SA^2+SW^2-3*SA*SW)-SW*(15*SA^2-8*SA*SW+2*SW^2))*S^2-3*(6*R^2-SW)*SB*SC*SW^2 : : (barys)
= 3*X(110)+X(18331), X(125)-3*X(15561), X(147)+3*X(15035), X(399)+3*X(14850), 7*X(3526)-3*X(14849), 3*X(5642)-X(18332), 3*X(8724)+X(18332), X(9862)-5*X(15051), X(13188)+3*X(14643), 3*X(14643)-X(16278), X(16111)-3*X(21166)
= lies on these lines: {99, 113}, {110, 18331}, {114, 17702}, {115, 12900}, {125, 15561}, {141, 542}, {147, 15035}, {399, 14850}, {541, 2482}, {690, 6132}, {2782, 5972}, {3526, 14849}, {5181, 12177}, {5477, 32661}, {5642, 8724}, {6033, 16163}, {6721, 15359}, {6723, 15535}, {7835, 15920}, {9862, 15051}, {11005, 30714}, {13188, 14643}, {15545, 24981}, {16111, 21166}
= midpoint of X(i) and X(j) for these {i,j}: {99, 113}, {5181, 12177}, {5642, 8724}, {6033, 16163}, {11005, 30714}, {13188, 16278}, {15545, 24981}
= reflection of X(i) in X(j) for these (i,j): (115, 12900), (6699, 620), (15359, 6721), (15535, 6723)
= {X(13188), X(14643)}-harmonic conjugate of X(16278)
= [ 0.5271006176178449, -2.6163923863120730, 5.2087358489152710 ]
César Lozada
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