HYACINTHOS 28071

 

 [César Lozada]:

 

Let ABC be a triangle, P a point and A’B’C’ the cyclocevian triangle of P. Circles (ABB’) and (ACC’) meet again in A” and  B”, C” are defined cyclically. Then ABC and A”B”C” are perspective.

 

If P=u:v:w (trilinears) then the perspector (ABC, A”B”C”)=Q(P) is:

Q(P) = a*u*(b*v+c*w)*(2*(a*u+b*v+c*w)*u*v*w*cos(B)+(u^2*v^2-u^2*w^2+v^2*w^2)*b)*(2*(a*u+b*v+c*w)*u*v*w*cos(C)+c*(-u^2*v^2+v^2*w^2+u^2*w^2)) : :

 

ETC pairs (P,Q(P)): (2,4), (4,6), (7,57), (8,282), (20,3344), (69,1073), (189,1), (253,2), (329,3342), (1031,251), (1032,3), (1138,11071), (1294,11070), (2113,9506), (2998,18018), (5932,3352), (6650,9472), (7357,7121), (8046,1168), (8050,9262)

 

Q( X(1) ) = X(35)X(37) ∩ X(319)X(321)

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

= on lines: {35, 37}, {319, 321}, {502, 594}, {1825, 8736}, {2171, 2594}

= barycentric product X(i)*X(j) for these {i,j}: {1, 502}, {10, 267}, {37, 1029}, {321, 3444}

= barycentric quotient X(i)/X(j) for these (i,j): (10, 20932), (25, 2906), (31, 501), (37, 2895), (42, 191), (213, 1030), (267, 86), (502, 75), (661, 21192), (756, 21081), (1029, 274), (1402, 8614), (1824, 451)

= trilinear product X(i)*X(j) for these {i,j}: {6, 502}, {10, 3444}, {37, 267}, {42, 1029}

= trilinear quotient X(i)/X(j) for these (i,j): (6, 501), (10, 2895), (19, 2906), (37, 191), (42, 1030), (267, 81), (321, 20932), (502, 2), (523, 21192), (594, 21081), (1029, 86), (1400, 8614), (1826, 451)

= [ -0.7858530931055387, -1.0884426482088370, 4.7569031274853610 ]

 

Q( X(3) ) = X(216)X(3463) ∩ X(324)X(17035)

= (S^4-SB^2*SC^2)*(3*S^2+8*R^2*(2*R^2-SB-SW)-4*SC*SA+2*SB^2+SW^2)*(3*S^2+8*R^2*(2*R^2-SC-SW)-4*SA*SB+2*SC^2+SW^2) : : (barys)

= on lines: {216, 3463}, {324, 17035}

= barycentric product X(5)*X(3463)

= barycentric quotient X(51)/X(3462)

= [ -0.1087823432918819, -0.0991508197373822, 3.7595145924757430 ]

 

Q( X(6) ) = X(32)X(15372) ∩ X(39)X(1915)

= (3*S^2-4*SC*SA+2*SB^2+SW^2)*(3*S^2-4*SA*SB+2*SC^2+SW^2)*(SA^2-SW^2) : : (barys)

= on the cubic K354 and these lines: {32, 15372}, {39, 1915}, {1031, 7779}, {1916, 8856}, {4074, 7794}, {8041, 9482}

= barycentric product X(i)*X(j) for these {i,j}: {39, 1031}, {141, 14370}, {1964, 18834}

= barycentric quotient X(i)/X(j) for these (i,j): (32, 14885), (38, 20934), (39, 2896), (1031, 308), (1401, 17083), (1964, 16556)

= trilinear product X(i)*X(j) for these {i,j}: {38, 14370}, {1031, 1964}

= trilinear quotient X(i)/X(j) for these (i,j): (31, 14885), (38, 2896), (39, 16556), (141, 20934), (1031, 3112), (1964, 10329)

= [ 0.1198466297519837, 0.1811667436105911, 3.4599275225992080 ]

 

César Lozada