HYACINTHOS 28064

[César Lozada]:

 

Let ABC be a triangle, P a point and A’B’C’ the tangential triangle of ABC.

Denote A” = the symmedian (Lemoine) point of PBC and similarly B” and C”.

Then the lines A’A”, B’B” and C’C” concur. (Source: AoPS)

 

For P=u:v:w (trilinears), the point of concurrence Q(P) is:

Q(P) = a*(-a*(-a^2+b^2+c^2)*v*w+4*a*b*c*u^2+b*(a^2+3*c^2-b^2)*w*u+c*(a^2+3*b^2-c^2)*u*v) : : (trilinears)

 

If P lies on the circumcircle of ABC, then Q(P)=P

If P lies on the line at infinity of ABC, then Q(P)=X(3)

If P lies on the Euler line of ABC, then Q(P) lies on the same line. If OP/OH=t, then OQ/OH=4*t/((OH/R)^2*t^2+3)

 

ETC pairs (P,Q(P)): (1,999), (2,11284), (3,3), (4,25), (15,11485), (16,11486), (23,25), (40,6244), (186,3515), (782,3), (895,14908), (3146,3515), (5537,6244), (14094,110), (15035,15020), (15054,74), (15156,1114), (15157,1113)

 

Some others:

Q(X(5)) = EULER LINE INTERCEPT OF X(54)X(18874)

= a^2*(a^8+a^4*b^2*c^2-2*(b^2+c^2)*a^6+(b^2+c^2)*(2*b^4-9*b^2*c^2+2*c^4)*a^2-(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-c^2)^2) : : (barys)

= As a point on the Euler line, this center has Shinagawa coefficients (5*E-8*F, 11*E+8*F)

= on lines: {2, 3}, {54, 18874}, {74, 18551}, {110, 13364}, {143, 12316}, {195, 10095}, {323, 13451}, {399, 5946}, {567, 14845}, {1614, 15047}, {3066, 18445}, {5640, 15087}, {5890, 12308}, {5898, 7693}, {5943, 10540}, {5966, 12074}, {6688, 13339}, {10564, 13570}, {10620, 15053}, {11451, 15038}, {12307, 14128}, {13321, 15068}, {13363, 14157}, {14627, 18350}, {20193, 21230}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5899, 3), (2, 7545, 5899), (3, 18535, 15684), (5, 10096, 2), (5, 13163, 4), (5, 13595, 2070), (25, 5055, 3), (1344, 1345, 3526), (1656, 13861, 18378), (1656, 18378, 3), (3843, 6642, 3), (3851, 7506, 3), (5070, 7517, 3), (5071, 14002, 7502), (11284, 12083, 15694), (12083, 15694, 3)

= [ 0.4740342750494653, -0.3967361622349503, 3.6965429288165140 ]

 

Q(X(6)) = X(3)X(6) ∩ X(25)X(1383)

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

= 3*S^2*X(3)-4*SW^2*X(6) = 3*X(3)-4*cot(ω)^2*X(6)

= on lines: {2, 3793}, {3, 6}, {25, 1383}, {30, 1285}, {112, 1597}, {115, 14269}, {172, 7373}, {183, 12150}, {193, 8369}, {230, 5055}, {248, 3531}, {251, 5020}, {381, 7735}, {382, 5305}, {385, 11286}, {729, 3511}, {1003, 7766}, {1572, 10247}, {1598, 3172}, {1627, 16419}, {1657, 5286}, {1914, 6767}, {1992, 6390}, {2207, 10985}, {2548, 3054}, {2549, 15681}, {3295, 7031}, {3517, 8743}, {3524, 14930}, {3534, 15048}, {3619, 7767}, {3620, 7819}, {3630, 7795}, {3631, 14023}, {3731, 5266}, {3767, 3843}, {3815, 15694}, {3830, 5306}, {3851, 7745}, {3933, 11008}, {3972, 14614}, {5054, 7736}, {5073, 5254}, {5077, 14712}, {5275, 16857}, {5276, 16418}, {5277, 16863}, {5309, 15684}, {5319, 17800}, {5355, 15685}, {5359, 9909}, {5475, 19709}, {5708, 16780}, {6144, 7801}, {6660, 13192}, {6781, 11742}, {7618, 20583}, {7738, 15696}, {7739, 15689}, {7746, 18584}, {7753, 15703}, {7755, 18424}, {7774, 11288}, {7798, 15301}, {7804, 8667}, {7806, 11318}, {7813, 15534}, {7866, 20065}, {7879, 10583}, {7887, 20088}, {7897, 8366}, {8584, 11165}, {9300, 15701}, {9463, 11328}, {9755, 10788}, {10304, 14482}, {10311, 18535}, {11179, 20194}, {11284, 11580}, {11287, 16989}, {11343, 14996}, {13638, 13763}, {13644, 13758}, {14001, 20080}, {14535, 15271}, {16318, 18494}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 187, 15603), (6, 574, 9605), (39, 14075, 6), (187, 574, 5585), (187, 5024, 3), (187, 5585, 15655), (187, 15602, 8588), (574, 3053, 15655), (574, 5007, 6), (574, 15655, 3), (1384, 5024, 187), (2080, 5050, 3), (3053, 5585, 187), (3311, 3312, 576), (6221, 6398, 3098), (8375, 8376, 11173)

= [ 0.4746985003648581, 1.0950216402838760, 2.6634809615425310 ]

 

Q(X(13)) = X(3)X(13) ∩ X(62)X(18350)

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

= on lines: {3, 13}, {62, 18350}, {1598, 8737}, {3457, 11486}, {10217, 16463}, {11081, 11485}

= {X(13), X(11142)}-harmonic conjugate of X(3)

= [ 0.0362890810209239, 0.0417288540576883, 3.5950264686270910 ]

 

Q(X(14)) = X(3)X(14) ∩ X(61)X(18350)

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

= on lines: {3, 14}, {61, 18350}, {1598, 8738}, {3458, 11485}, {10218, 16464}, {11086, 11486}

= {X(14), X(11141)}-harmonic conjugate of X(3)

= [ 3.7682185486635580, 6.9015504072075690, -2.8765097455425100 ]

 

Q(X(20)) = anticomplement of X(1596)

= a^2*(a^8+16*a^4*b^2*c^2-2*(b^2+c^2)*a^6+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2-(b^4+6*b^2*c^2+c^4)*(b^2-c^2)^2) : : (barys)

= As a point on the Euler line, this center has Shinagawa coefficients (E-F, -2*E+F)

= 3*X(5085)-2*X(19136), 3*X(6090)-2*X(18451), 2*X(8263)-3*X(10519), 3*X(15035)-2*X(20772)

= on lines: {2, 3}, {64, 5562}, {74, 3565}, {98, 20187}, {99, 1294}, {110, 11820}, {112, 1033}, {155, 10575}, {159, 2935}, {185, 12160}, {394, 6000}, {511, 10602}, {691, 2693}, {841, 10420}, {999, 3100}, {1062, 1398}, {1147, 14641}, {1181, 13346}, {1204, 17834}, {1292, 1295}, {1296, 1297}, {1350, 2393}, {1351, 5890}, {1384, 10313}, {1473, 7171}, {1565, 4329}, {1578, 11473}, {1579, 11474}, {1619, 15311}, {1993, 15072}, {2691, 2694}, {2696, 2697}, {2734, 2737}, {2794, 2936}, {2979, 13445}, {3098, 14913}, {3167, 11456}, {3184, 15259}, {3295, 4296}, {3304, 9643}, {3357, 15644}, {3426, 15066}, {3527, 15043}, {3587, 7085}, {3695, 7219}, {3796, 11430}, {4293, 16541}, {5050, 15033}, {5085, 19136}, {5422, 20791}, {5891, 11472}, {5925, 9914}, {6090, 14915}, {6221, 11417}, {6241, 12164}, {6361, 12410}, {6398, 11418}, {6560, 19006}, {6561, 19005}, {7373, 9538}, {8192, 18481}, {8263, 10519}, {8717, 18475}, {8718, 9707}, {8778, 10316}, {8780, 14157}, {9729, 10982}, {9730, 9777}, {9937, 11750}, {10037, 10483}, {10574, 11432}, {10601, 16836}, {10625, 12163}, {10831, 15338}, {10832, 15326}, {10984, 11425}, {11381, 17814}, {11402, 13352}, {11406, 15941}, {11441, 12279}, {11457, 12429}, {11477, 14831}, {12017, 19121}, {12058, 13754}, {12111, 13093}, {12118, 12166}, {12256, 12303}, {12257, 12304}, {12302, 13171}, {12412, 20127}, {13142, 18916}, {13491, 16266}, {15030, 17811}, {15035, 20772}

= anticomplement of X(1596)

= circumperp conjugate of X(468)

= X(25)-of-ABC-X3 reflections triangle

= X(1368)-of-anti-Euler triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 6642), (3, 1593, 7395), (3, 1597, 2), (3, 5073, 7506), (3, 12083, 14070), (22, 2071, 3), (22, 7396, 5020), (22, 11413, 2071), (186, 12082, 9909), (382, 6642, 5198), (548, 7526, 3), (1593, 7484, 9818), (1995, 3543, 18535), (2041, 2042, 1906), (3146, 17928, 1598), (3528, 14865, 7509)

= [ 14.1111305253582500, 13.2043850838333400, -12.0136623570655900 ]

 

Q( X(25) ) = EULER LINE INTERCEPT OF X(1974)X(6391)

= SB*SC*(SB+SC)*(4*S^2*(6*R^2-SW)-SA^2*SW) : : (barys)

= on lines: {2, 3}, {1974, 6391}, {5544, 19124}, {15531, 19118}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 1995, 1598), (25, 4232, 3517), (25, 6353, 9909), (5020, 9909, 1368)

= [ -0.0115895959991488, -0.8810789452498955, 4.2559912575416070 ]

 

César Lozada