HYACINTHOS 29189

[Antreas P. Hatzipolakis]:

 

 

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

Denote:

A", B", C" = the antipodes of A', B', C' in the pedal circle of P ( = circumcircle of A'B'C'), resp.

Na, Nb, Nc = the NPC centers of A"BC, B"CA, C"AB, resp.

ABC, NaNbNc are orthologic.

Orthologic centers in terms of P?

 

 

 

[César Lozada]:

 

 

For P=x:y:z (barys) the orthologic centers are:

 

Qa = A->Na = a^2*(2*a^2*x*y*z+(y+z)*a^2*y*z+(b^2*z+c^2*y)*x^2)/(-4*(y^2+2*y*z+z^2)*(-a^2+b^2+c^2)*a^6*y^2*z^2+(12*(a^2-b^2)*b^2*z^2+12*(a^2-c^2)*c^2*y^2+(5*a^4-13*b^4+2*b^2*c^2-13*c^4+8*(b^2+c^2)*a^2)*y*z)*a^4*x^2*y*z+2*(y+z)*(2*(a^2-b^2)*b^2*z^2+2*(a^2-c^2)*c^2*y^2+(5*a^4-2*b^4-2*c^4-3*(b^2+c^2)*a^2)*y*z)*a^4*x*y*z+2*(-(-a^2+b^2+c^2)*b^4*z^3-(-a^2+b^2+c^2)*c^4*y^3+(3*a^4-2*b^4-b^2*c^2-3*c^4-(b^2-6*c^2)*a^2)*b^2*y*z^2+(3*a^4-3*b^4-b^2*c^2-2*c^4+(6*b^2-c^2)*a^2)*c^2*y^2*z)*a^2*x^3+((a^2+b^2-c^2)*(a^2-b^2+c^2)*b^4*z^2+2*(3*a^4-(b^2-c^2)^2)*b^2*c^2*y*z+(a^2+b^2-c^2)*(a^2-b^2+c^2)*c^4*y^2)*x^4) : :

 

Qn = Na->A = a^2*((2*a^4-3*(b^2+c^2)*a^2+(b^2-c^2)^2)*y*z-(-a^2+b^2+c^2)*(b^2*z+c^2*y)*x) : :

 

If P ∈ {circumcircle} ∪ {Linf}  then Qa(P)=Isogonal(P)

 

If P ∈ {circumcircle} then Qn(P)=Isogonal(P). otherwise Qn(P)= Midpoint(Isogonal(P), O)

 

ETC pairs (P,Qn): (1,1385), (2,182), (3,5), (4,3), (5,10610), (6,549), (7,32613), (8,32612), (13,13350), (14,13349), (15,6771), (16,6774), (20,3357), (24,12359), (28,31837), (36,12619), (54,140), (55,31657), (56,5690), (57,31658), (58,6684), (59,6713), (60,31659), (64,550), (65,5428), (66,7502), (67,7575), (69,6644), (70,1658), (76,13335), (80,23961), (83,13334), (84,3579), (93,13367), (94,22463), (95,5892), (96,389), (186,125), (249,6036), (250,6699), (252,12006), (253,11202), (254,1147), (262,5092), (264,18475), (265,15646), (267,22937), (376,11472), (631,15805), (670,5926), (685,8552), (847,12038), (892,9126), (943,9940), (947,1125), (963,22791), (1000,10269), (1131,12974), (1132,12975), (1138,110), (1157,12026), (1166,32348), (1173,3530), (1177,15122), (1179,5447), (1263,6150), (1320,18857), (1389,13624), (1498,20329), (1676,8161), (1677,8160), (2065,620), (2218,24475), (3065,10225), (3147,19360), (3224,32448), (3296,10267), (3346,6759), (3417,10), (3424,3098), (3425,141), (3426,8703), (3431,2), (3432,21230), (3445,1483), (3446,1484), (3447,10264), (3459,54), (3524,5544), (3527,15712), (3531,17504), (3532,3627), (3577,17502), (4846,18570), (5468,9175), (5481,3589), (5505,18579), (5553,26285), (5879,2883), (5900,2070), (6188,10540), (6662,5944) , (7607,575), (7608,20190), (7612,6) , (8770,1353), (8884,1216), (9302,2080) , (10155,12017), (10305,11248), (10308,31663), (10411,1116), (10419,5972), (10623,946), (11170,21163), (11270,4), (11564,18571), (11653,16324), (11738,376), (11815,32142), (11816,10627), (12028,14708), (13139,15045), (13452,20), (13472,631), (13573,13289), (13574,12584), (13575,15577), (14364,14649), (14380,32162), (14458,14810), (14483,12100), (14484,17508), (14487,14891), (14491,3524), (14494,5085), (14495,21167), (14497,3576), (14528,632), (15318,10282), (15364,15361), (15378,6712), (15379,6718), (15380,6710), (15381,3035), (15386,6711), (15395,31379), (15396,22104) and maybe more

 

 

Qa(X(1))= X(1)X(5927) ∩ X(4)X(9949)

= a*(a^3-(b-5*c)*a^2-(b+c)^2*a+(b^2-c^2)*(b+5*c))*(a^3+(5*b-c)*a^2-(b+c)^2*a-(b^2-c^2)*(5*b+c)) : : (barys)

= lies on the Feuerbach hyperbola and these lines: {1, 5927}, {4, 9949}, {7, 3832}, {8, 9580}, {9, 3983}, {21, 5438}, {72, 4900}, {80, 9589}, {84, 19541}, {90, 10860}, {104, 10864}, {226, 5558}, {950, 7320}, {1000, 5881}, {1155, 7285}, {1156, 5128}, {1172, 16670}, {1320, 11523}, {1728, 3065}, {1864, 5665}, {2320, 5436}, {2335, 16676}, {2346, 10384}, {3062, 5729}, {3296, 5714}, {3427, 31673}, {3577, 9856}, {3586, 5559}, {5225, 6601}, {5557, 9612}, {5560, 30286}, {7091, 17604}, {7160, 9947}, {7655, 23838}, {8581, 10390}, {9948, 10309}, {10398, 31507}

= [ 5.1099929085129870, 6.5536231920839200, -3.2549175319258670 ]

 

Qa(X(2)): not interesting

 

Qa(X(3)) = X(13450)

 

Qa(X(4)) = (name pending)

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

= lies on the line {1593, 18439}

= [ 127.1631507824772000, 40.5787608368357100, -83.1430087662760000 ]

 

César Lozada