HYACINTHOS 29211

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the cevian triangle of G and P a point.

Denote: 

A", B", C" = the reflections of A, B, C in A", B", C", resp.
(ie A"B"C" is the antimedial triangle of ABC)

Pa, Pb, Pc = the isogonal conjugates of P wrt triangles A"BC, B"CA, C"AB, resp.

Which is the locus of P such that ABC, PaPbPc are cyclologic?
The entire plane? (are they circumcyclologic for all P's ?)


[César Lozada]:

 

Locus for both questions = The entire plane.

 

Assume P = x: y : z (barys)

 

Qa = (ABC à  PaPbPc) =

a^2/(((a^2-2*b^2+2*c^2)*y-(a^2+2*b^2-2*c^2)*z)*a^2*y*z+((a^4+b^4-c^4-(2*b^2-c^2)*a^2)*y-(a^4-b^4+c^4+(b^2-2*c^2)*a^2)*z)*x^2+((a^4-2*(b^2-c^2)*a^2+(b^2-c^2)*b^2)*y^2-(3*a^2-b^2-c^2)*(b^2-c^2)*y*z-(a^4+2*(b^2-c^2)*a^2-(b^2-c^2)*c^2)*z^2)*x) : :

 

Qp = (PaPbPc à ABC) =

(a^2*y*z+(a^2-b^2-c^2)*x^2+((a^2-b^2)*y+(a^2-c^2)*z)*x)/(-(-a^2+b^2+c^2)*a^2*y*z+((-a^2+b^2+c^2)^2-b^2*c^2)*x^2+(((a^2-b^2)^2-c^2*a^2)*y+((a^2-c^2)^2-a^2*b^2)*z)*x) : :

= Isogonal( InverseInCircumcircle( AntigonalConjugate( P ) ) )

 

Some ETC pairs (P, Qa(P)): (3,925), (6,99), (30,110), (54,930), (64,107), (65,100), (66,112), (67,691), (68,13398), (69,3565), (71,1305), (72,13397), (74,476), (80,901), (98,691), (99,842), (100,2687), (101,2688), (105,2691), (106,2692), (107,2693), (109,2695), (110,477), (111,2696), (112,2697), (115,691), (146,1304), (147,6037), (148,9150), (149,898), (150,901), (290,805), (316,99), (511,99), (512,98), (513,104), (514,103), (515,109), (516,101), (517,100), (518,1292), (520,1294), (521,1295), (522,102), (523,74), (524,1296), (525,1297), (526,477), (527,28291), (528,2742), (536,28474), (537,28520), (539,20185), (542,691), (543,2709), (545,28293), (621,30253), (622,30252), (671,691), (690,842), (691,98), (695,689), (698,30254), (726,28469), (740,6010), (752,28467), (758,6011), (804,2698), (812,12032), (814,29009), (824,28844), (826,29011), (879,1297), (891,29348), (900,953), (918,28838), (924,1300), (925,32710), (926,2724), (928,2723), (930,14979), (935,1297), (952,901), (971,934)

 

Some ETC pairs (P,Qp(P)): (1,33599), (2,33601), (3,5962), (5,11584), (6,316), (13, 11601), (14,11600),  (54,19552), (65,5080), (66,5523), (67,671), (68,16172), (69,5203), (79,19658), (80,80), (115,3448), (186,5964), (316,6), (403,15242), (671,67) , (1177,11605), (1263,33565), (1320,17101)

For P ∈ {Linf}, Qp( P ) = X(4).        

 

Some others related centers:

Qa( X(1) ) = TRILINEAR POLE OF LINE X(6)X(1731)

= a*(a-b)*(a-c)*(a^3-b*a^2-b^2*a+(b^2-c^2)*(b-c))*(a^3-c*a^2-c^2*a+(b^2-c^2)*(b-c)) : : (barys)

= lies on the circumcircle and these lines: {3, 14987}, {46, 15530}, {104, 411}, {105, 29681}, {106, 21842}, {109, 13589}, {915, 7412}, {1018, 29115}, {1897, 30250}, {3799, 29337}, {4242, 26704}, {6099, 14513}, {7424, 12030}

= reflection of X(14987) in X(3)

= circumperp conjugate of X(14987)

= antipode of X(14987) in the circumcircle

= trilinear pole of the line X(6)X(1731)

= barycentric quotient X(i)/X(j) for these (i,j): (100, 33113), (101, 22836) et al.

= trilinear quotient X(i)/X(j) for these (i,j): (100, 22836), (190, 33113) et al.

= X(12092)-of-1st circumperp triangle

= X(14103)-of-excentral triangle

= X(14987)-of-ABC-X3 reflections triangle

= X(22751)-of-2nd circumperp triangle

= [ 5.9576839938916120, 11.9575711801838800, -7.3873543323236600 ]

 

Qa( X(2) ) = TRILINEAR POLE OF LINE X(6)X(9716)

= a^2*(2*a^2-7*b^2+2*c^2)*(2*a^2+2*b^2-7*c^2)*(a^2-c^2)*(a^2-b^2) : : (barys)

= lies on the circumcircle and these lines: {98, 3534}, {111, 7492}, {376, 13530}, {843, 8586}, {2374, 5094}, {2770, 10989}, {4563, 6082}, {5966, 15744}, {5970, 10631}, {11634, 11636}

= trilinear pole of the line X(6)X(9716) et al.

= barycentric quotient X(110)/X(15534) et al.

= trilinear quotient X(662)/X(15534)

= [ 8.2809672331963120, 12.9824119221260700, -9.1691440333473540 ]

 

Qa( X(5) ) = ISOGONAL CONJUGATE OF X(13152)

= (SB+SC)*(SA-SB)*(SA-SC)*(4*SB-R^2-2*SW)*(4*SC-R^2-2*SW) : : (barys)

= lies on the circumcircle and these lines: {74, 12291}, {2383, 3520}, {5966, 6636}

= anticomplement of the complementary conjugate of X(13152)

= complement of the anticomplementary conjugate of X(13152)

= isogonal conjugate of X(13152)

= trilinear pole of the line {6, 22462}

= [ 13.1934041632953300, 12.2492965187843100, -10.9288811833872200 ]

 

Qa( X(20) ) = TRILINEAR POLE OF LINE X(6)X(3520)

= SB*SC*(SB+SC)*(SA-SB)*(SA-SC)*(S^2-5*SA*SB)*(S^2-5*SA*SC) : : (barys)

= lies on the circumcircle and these lines: {4, 20480}, {74, 6759}, {107, 5502}, {1294, 1657}, {2693, 6760}, {2697, 30745}, {5896, 14379}, {12092, 30510}, {12107, 18401}

= trilinear pole of the line {6, 3520}

= barycentric product X(648)*X(11270) et al.

= barycentric quotient X(i)/X(j) for these (i,j): (112, 382), (162, 14212) et al.

= trilinear product X(162)*X(11270) et al.

= trilinear quotient X(i)/X(j) for these (i,j): (162, 382), (648, 14212) et al.

= [ 0.6605010031003141, -0.7215269512657981, 3.8353365236605570 ]

 

Qp( X(20) ) = ANTIGONAL CONJUGATE OF X(11589)

= SA*(S^2-3*SB*SC)*(S^2+(40*R^2-12*SW+5*SB)*SB)*(S^2+(40*R^2-12*SW+5*SC)*SC) : : (barys)

= lies on the cubic K427 and this line {382, 1853}

= antigonal conjugate of X(11589)

= [ -1.1572130338144530, -1.4458547122125750, 5.1757391444305250 ]

 

César Lozada