Let ABC be a triangle and P a point.
Denote:
A', B', C' = the midpoints of AP, BP, CP, resp.
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
The perpendicular from A' to CA intersects again (Nb) at Ab
The perpendicular from A' to AB intersects again (Nc) at Ac
The perpendicular from B' to AB intersects again (Nc) at Bc
The perpendicular from B' to BC intersects again (Na) at Ba
The perpendicular from C' to BC intersects again (Na) at Ca
The perpendicular from C' to CA intersects again (Nb) at Bc
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
Which is the locus of P such that
1. ABC, MaMbMc are orthologic?
The entire plane?
2. The perpendicular bisectors of AbAc, BcBa, CaCb are concurrent?.[César Lozada]:
1) The entire plane.
For P=x:y:z (barys), the orthologic center ABC->MaMbMc is:
Qa = x/(-3*(-a^2+b^2+c^2)*x^2+(3*a^2-3*b^2-c^2)*x*y+(3*a^2-b^2-3*c^2)*x*z+2*a^2*y*z) : :
and the orthologic center MaMbMc->ABC is:
Qm = - 4*(x+y+z)*x*y*z*SB*SC-(SB*c^2*y^2+SC*b^2*z^2)*x^2+a^4*y^2*z^2 : :
If P lies on the circumcircle, Qa(P)=X(4). If P lies in the infinity, Qa(P)=P.
If P lies on the circumcircle, Qm(P)=midpoint(P, X(382)).
Other ETC pairs (P, Qa): (1,7319), (4,4), (6,66), (80,7), (253,1249), (265,4846), (671,18842), (3062,7), (3426,4846), (3577,3427), (10152,1249), (11744,66)
(P, Qm(P)): (1,4), (2,15687), (3,18383), (4,5), (30,32417), (57,18516), (67,20301), (80,11), (84,10525), (3062,2550)
Some related centers:
Qa( X(2) ) = ISOGONAL CONJUGATE OF X(15655)
= (7*a^2-11*b^2+7*c^2)*(7*a^2+7*b^2-11*c^2) : : (barys)
= lies on the Kiepert hyperbola and these lines: {4, 8584}, {98, 15682}, {115, 10153}, {376, 7607}, {631, 10185}, {1992, 17503}, {2996, 8352}, {3424, 3830}, {3545, 7608}, {3845, 14484}, {5254, 18843}, {5286, 18844}, {5395, 11317}, {5485, 15533}, {7612, 11001}, {11668, 15719}
= reflection of X(10153) in X(115)
= antigonal conjugate of X(10153)
= antitomic conjugate of X(10153)
= isogonal conjugate of X(15655)
= isotomic conjugate of the anticomplement of X(15534)
= antipode of X(10153) in the Kiepert hyperbola
= [ 3.6067375476142880, 5.9847091169972360, -2.1672437749126080 ]
Qa( X(3) ) = X(5)X(14528) ∩ X(6)X(546)
= (-a^2+b^2+c^2)*(3*a^4-2*(3*b^2-2*c^2)*a^2+3*(b^2-c^2)^2)*(3*a^4+2*(2*b^2-3*c^2)*a^2+3*(b^2-c^2)^2) : : (barys)
= SA*(S^2-5*SB^2)*(S^2-5*SC^2) : : (barys)
= 7*X(3090)-3*X(25712)
= lies on the Jerabek hyperbola and these lines: {5, 14528}, {6, 546}, {30, 3532}, {54, 3091}, {64, 3627}, {66, 18383}, {67, 18569}, {68, 13851}, {69, 9927}, {70, 18394}, {74, 3146}, {895, 15083}, {1352, 13622}, {1899, 3521}, {3090, 3431}, {3426, 5076}, {3519, 18404}, {3529, 11270}, {3839, 10116}, {5486, 18553}, {10113, 11744}, {10293, 31725}, {10297, 15316}, {10594, 18532}, {12102, 22334}, {13754, 15077}, {14542, 18390}, {15740, 18918}, {16625, 18376}, {17538, 20421}, {21400, 25738}
= isogonal conjugate of Qa( X(3) )*
= [ -0.8915870954373679, -1.1156935179158950, 4.8245694229741680 ]
Qa( X(3) )* = ISOGONAL CONJUGATE OF Qa( X(3) )
= a^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(3*a^4-6*(b^2+c^2)*a^2+3*b^4+4*b^2*c^2+3*c^4) : : (barys)
= SB*SC*(SB+SC)*(S^2-5*SA^2) : : (barys)
= As a point on the Euler line, this center has Shinagawa coefficients (-6*F, E+6*F)
= lies on these lines: {2, 3}, {54, 9786}, {64, 11468}, {74, 1498}, {110, 12163}, {112, 5023}, {154, 6241}, {155, 11449}, {184, 13382}, {185, 9707}, {187, 8743}, {232, 5206}, {340, 9723}, {1151, 10881}, {1152, 10880}, {1181, 11464}, {1192, 5890}, {1199, 3431}, {1204, 10282}, {1350, 19128}, {1495, 3357}, {1511, 13148}, {1614, 1620}, {1829, 17502}, {1843, 17508}, {1870, 5204}, {1968, 15513}, {1974, 14810}, {1986, 11412}, {1990, 8553}, {1993, 12038}, {2207, 5210}, {2351, 5963}, {2904, 16879}, {2931, 9938}, {3043, 15040}, {3092, 6411}, {3093, 6412}, {3567, 11425}, {3580, 12118}, {3581, 16266}, {5013, 10312}, {5085, 6403}, {5217, 6198}, {5237, 8740}, {5238, 8739}, {5410, 6450}, {5411, 6449}, {5621, 15581}, {5702, 8573}, {6030, 15086}, {6759, 21663}, {7592, 11438}, {7689, 11441}, {8541, 20190}, {8744, 8778}, {8907, 12893}, {8960, 9682}, {10193, 13419}, {10606, 12290}, {10619, 23358}, {10632, 11481}, {10633, 11480}, {10990, 13289}, {11204, 11381}, {11270, 12315}, {11363, 31663}, {11440, 18451}, {11550, 25563}, {12112, 13093}, {12278, 14852}, {12279, 12292}, {12383, 12429}, {13481, 19189}, {13558, 16080}, {15036, 15472}, {15578, 20987}, {16655, 23328}, {17845, 25739}, {18396, 26917}, {18445, 32171}, {18474, 20191}, {19467, 26879}, {20427, 32111}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7387, 2071), (4, 3515, 24), (14865, 23040, 11410)
= [ 4.3864339295720590, 3.5053424832122090, -0.8106190508881003 ]
Qa( X(5) ) = X(3)X(1487) ∩ X(4)X(1493)
= (7*R^2-3*SC-SW)*(7*R^2-3*SB-SW)*(S^2+SB*SC) : : (barys)
= lies on these lines: {3, 1487}, {4, 1493}, {20, 3459}, {3853, 15619}, {17507, 20424}
= [ -6.2066991061248400, -5.0374033377571830, 9.9927279185662030 ]
Qm( X(5) ) = X(4)X(137) ∩ X(5)X(27684)
= 3*S^4-(R^2*(12*R^2+5*SA-12*SW)-2*SA^2+13*SB*SC+3*SW^2)*S^2-(R^2*(28*R^2-17*SW)+SW^2)*SB*SC : : (barys)
= 3*X(381)-X(30484)
= lies on these lines: {4, 137}, {5, 27684}, {30, 13856}, {128, 24573}, {381, 30484}, {539, 23337}, {546, 20413}, {3853, 5893}, {8146, 28341}, {15307, 31376}, {15619, 17507}, {18807, 28237}, {20120, 25150}
= reflection of X(i) in X(j) for these (i,j): (13856, 31879), (18016, 23338)
= [ -8.7400154720944610, -11.3742123610257500, 15.5489724881996500 ]
2) The entire plane.
For P=x:y:z (barys), they concur at:
Q = x*((-4*a^6*c^2+2*c^2*(7*c^2+6*b^2)*a^4-4*c^2*(4*c^4+5*b^2*c^2+3*b^4)*a^2+2*c^2*(8*b^2*c^4+2*b^6+3*c^6+3*b^4*c^2))*z*y^3*x^4+(-8*a^6*c^2+8*c^2*(c^2+3*b^2)*a^4-4*c^2*(6*b^4-c^4)*a^2+4*(b^2-c^2)*c^2*(c^4+2*b^4))*z*y^4*x^3+(-4*a^6*c^2+6*c^2*(2*b^2-c^2)*a^4-4*(b^2-c^2)*a^2*c^2*(3*b^2-2*c^2)+2*(b^2-c^2)^2*c^2*(2*b^2-3*c^2))*z*y^5*x^2+(3*a^8+(-3*c^2-9*b^2)*a^6+(-3*c^4+10*b^2*c^2+9*b^4)*a^4+(3*c^6-3*b^6-5*b^2*c^4-35*b^4*c^2)*a^2+4*(b^2-c^2)*b^2*c^2*(-5*c^2+7*b^2))*x^3*y^3*z^2+(-a^8+(-9*c^2+5*b^2)*a^6-9*(b^2-c^2)^2*a^4+(b^2-c^2)^2*a^2*(7*b^2-3*c^2)-2*(b^2-c^2)^4)*x*y^5*z^2-8*a^6*y^3*z^5*b^2+(8*a^8+(-8*c^2-8*b^2)*a^6)*z^4*y^4-8*a^6*y^5*z^3*c^2+(12*a^8+(-18*b^2-18*c^2)*a^6+(12*b^2*c^2-12*c^4-12*b^4)*a^4+30*(b^2-c^2)^2*(b^2+c^2)*a^2+(b^2-c^2)^2*(12*b^2*c^2-12*c^4-12*b^4))*z^3*y^3*x^2+(-a^8+(-9*b^2+5*c^2)*a^6-9*(b^2-c^2)^2*a^4-(b^2-c^2)^2*a^2*(3*b^2-7*c^2)-2*(b^2-c^2)^4)*x*y^2*z^5+(2*a^8+(-7*c^2-7*b^2)*a^6+(9*b^4+9*c^4+26*b^2*c^2)*a^4-(b^2+c^2)*(5*c^4+34*b^2*c^2+5*b^4)*a^2+20*b^2*c^6+c^8+6*b^4*c^4+b^8+20*b^6*c^2)*x^4*y^2*z^2+(3*a^8+(-9*c^2-3*b^2)*a^6+(-3*b^4+9*c^4+10*b^2*c^2)*a^4+(-3*c^6-35*b^2*c^4-5*b^4*c^2+3*b^6)*a^2+4*(b^2-c^2)*b^2*c^2*(-7*c^2+5*b^2))*z^3*y^2*x^3+(9*a^8+(-19*b^2-9*c^2)*a^6+5*(b^2-c^2)^2*a^4+(b^2-c^2)^2*a^2*(11*b^2+c^2)-6*(b^2-c^2)^4)*z^3*y^4*x+(9*a^8+(-19*c^2-9*b^2)*a^6+5*(b^2-c^2)^2*a^4+(b^2-c^2)^2*a^2*(b^2+11*c^2)-6*(b^2-c^2)^4)*z^4*y^3*x+(-4*a^6*b^2-6*b^2*(b^2-2*c^2)*a^4-4*(b^2-c^2)*a^2*b^2*(2*b^2-3*c^2)-2*(b^2-c^2)^2*b^2*(3*b^2-2*c^2))*z^5*y*x^2+(-4*a^2*b^6+4*b^6*(b^2+c^2))*x^4*z^4+(-4*a^2*b^6-4*(b^2-c^2)*b^6)*x^3*z^5+(-4*a^2*c^6+4*(b^2-c^2)*c^6)*x^3*y^5+(-4*a^2*c^6+4*c^6*(b^2+c^2))*x^4*y^4+((-5*c^2+3*b^2)*a^6-(b^2-c^2)*a^4*(7*c^2+9*b^2)+(b^4-c^4)*a^2*(-13*c^2+9*b^2)+(b^2-c^2)^2*(-3*b^4+10*b^2*c^2-15*c^4))*z^2*y^4*x^2+((3*c^2-5*b^2)*a^6+(b^2-c^2)*a^4*(9*c^2+7*b^2)+(b^4-c^4)*a^2*(-9*c^2+13*b^2)+(b^2-c^2)^2*(-15*b^4+10*b^2*c^2-3*c^4))*z^4*y^2*x^2+(-8*a^6*b^2+8*b^2*(3*c^2+b^2)*a^4+4*b^2*(-6*c^4+b^4)*a^2-4*(b^2-c^2)*b^2*(2*c^4+b^4))*z^4*y*x^3+(-4*a^6*b^2+2*b^2*(7*b^2+6*c^2)*a^4-4*b^2*(3*c^4+5*b^2*c^2+4*b^4)*a^2+2*b^2*(3*b^2*c^4+3*b^6+2*c^6+8*b^4*c^2))*z^3*y*x^4) : :
For P on the circumcircle, Q(P)=midpoint(P, X(382)). For P at the infinity, Q(P) = P.
ETC pairs (P, Q(P) ): (4, 5)
Some related centers:
Q (X(1) ) = X(5)X(519) ∩ X(7)X(8)
= 2*a^4-3*(b+c)*a^3+10*a^2*b*c+(b+c)*(3*b^2-8*b*c+3*c^2)*a-2*(b^2-c^2)^2 : : (barys)
= X(3621)+3*X(25568), 3*X(3679)-X(12513), 5*X(4668)-X(6762), 3*X(4669)-X(24391), 3*X(4677)+X(11523), 7*X(4678)-3*X(24477), 3*X(5587)-X(10912), X(7982)-3*X(11236)
= lies on these lines: {1, 5123}, {5, 519}, {7, 8}, {10, 6691}, {145, 11376}, {355, 3880}, {474, 3679}, {517, 32159}, {528, 12640}, {529, 11362}, {946, 5854}, {952, 10915}, {960, 12647}, {1012, 3913}, {1210, 3036}, {1317, 27385}, {1319, 17566}, {1387, 3244}, {1482, 22835}, {1837, 12648}, {2098, 5087}, {2802, 18480}, {3057, 5046}, {3241, 6931}, {3337, 4668}, {3621, 25568}, {3625, 5855}, {3632, 10827}, {3633, 23708}, {3742, 5554}, {3811, 12645}, {3838, 9578}, {3893, 5086}, {4134, 15862}, {4297, 32157}, {4669, 24391}, {4677, 11523}, {4678, 24477}, {4861, 7504}, {5048, 11681}, {5258, 19525}, {5587, 10912}, {5795, 15254}, {5837, 15481}, {5844, 21077}, {6735, 10944}, {6738, 15570}, {6871, 31145}, {7483, 10039}, {7982, 11236}, {7991, 28534}, {8256, 10106}, {8715, 28204}, {9037, 31785}, {9956, 22837}, {11567, 22836}, {12641, 13271}, {12672, 12751}, {13463, 19925}, {15888, 25962}, {20323, 25005}, {21627, 32426}
= midpoint of X(i) and X(j) for these {i,j}: {8, 32049}, {3632, 12635}, {3811, 12645}, {3913, 5881}, {12641, 13271}
= reflection of X(i) in X(j) for these (i,j): (4297, 32157), (11260, 10), (13463, 19925), (22837, 9956)
= {X(8), X(5252)}-harmonic conjugate of X(5836)
= [ -4.2187283215198470, -0.6362133417539666, 6.0282252515155130 ]
Q (X(2) ) = X(381)X(7620) ∩ X(3734)X(16509)
= 126*S^4+3*(3*SA+SW)*(15*SA-14*SW)*S^2-4*(3*SA+SW)*(3*SA-2*SW)*SW^2 : : (barys)
= lies on these lines: {381, 7620}, {3734, 16509}, {5485, 18907}
= [ -28.1524077663532100, 9.2504991662147700, 10.2298917205371700 ]
Q (X(3) ) = X(4)X(52) ∩ X(125)X(1147)
= SA*((16*R^2-2*SA-2*SW)*S^2+(SB+SC)*(10*R^4-(7*SA+11*SW)*R^2+2*SA^2-2*SB*SC+2*SW^2)) : : (barys)
= lies on these lines: {4, 52}, {5, 32166}, {125, 1147}, {155, 10255}, {539, 18281}, {1899, 12038}, {3357, 17702}, {3448, 12118}, {3564, 10224}, {5449, 9306}, {5504, 23294}, {6696, 11250}, {12163, 18565}, {13383, 32145}, {15316, 19477}
= reflection of X(32145) in X(13383)
= {X(68), X(11442)}-harmonic conjugate of X(9927)
= [ 3.8866985799126010, 6.4562459709753850, -2.6229051502659330 ]
César Lozada
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