HYACINTHOS 27196

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and F the Feuerbach point.

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.
Bc, Ba = the orthogonal projections of B on CI, AI, resp.
Ca, Cb = the orthogonal projections of C on AI, BI, resp..

(Na), (Nb), (Nc) = the NPCs of FAbAc, FBcBa, FCaCb, resp.

1. (Na), (Nb), (Nc) are concurrent at the midpoint of IF

 

2. ABC, NaNbNc are perspective (homothetic).
Perspector (homothetic center) = U on the IF line such that UI / UF = -2  (?)

3. ABC, NaNbNc are orthologic.
Which is the orthologic center (NaNbNc, ABC) = orthocenter of NaNbNc?

[César Lozada]:

 

1)      X(1387)

2)

Q = {X(1),X(11)}-harmonic conjugate of X(80)

= a^4-(b+c)*a^3-(2*b-c)*(b-2*c)* a^2+(b^2-c^2)*(b-c)*a+(b^2-c^ 2)^2 : : (barys)

= = X(1)+2*X(11), 2*X(1)+X(80), 5*X(1)-2*X(1317), X(1)-4*X(1387), 4*X(1)-X(7972), 5*X(1)+X(9897), 5*X(1)+4*X(12019), 7*X(1)-4*X(12735), 2*X(5)+X(12737), 4*X(5)-X(12751), 4*X(11)-X(80), 5*X(11)+X(1317), X(11)+2*X(1387), 8*X(11)+X(7972), 10*X(11)-X(9897), 5*X(11)-2*X(12019), 7*X(11)+2*X(12735), 5*X(80)+4*X(1317), X(80)+8*X(1387), 2*X(80)+X(7972), 5*X(80)-2*X(9897), 5*X(80)-8*X(12019), 7*X(80)+8*X(12735), 2*X(119)+X(6264), 2*X(119)-5*X(8227), 4*X(496)-X(10073), 2*X(496)+X(12740), X(1317)-10*X(1387), 8*X(1317)-5*X(7972), 2*X(1317)+X(9897), X(1317)+2*X(12019), 7*X(1317)-10*X(12735), 16*X(1387)-X(7972), 5*X(1387)+X(12019), 7*X(1387)-X(12735), X(1484)+2*X(5901), 2*X(1484)+X(6265), X(5660)-4*X(5886), 4*X(5901)-X(6265), X(6264)+5*X(8227), X(6326)-7*X(9624), X(6326)-4*X(11729), 5*X(7972)+4*X(9897), 7*X(7972)-16*X(12735), X(7993)+5*X(15017), 7*X(9624)-4*X(11729), X(9897)-4*X(12019), X(10073)+2*X(12740), 7*X(12019)+5*X(12735), 2*X(12737)+X(12751)

= on lines: {1, 5}, {2, 2802}, {3, 14217}, {4, 11715}, {8, 6702}, {10, 1320}, {35, 6940}, {36, 516}, {40, 5442}, {55, 5444}, {79, 104}, {100, 1125}, {106, 3120}, {145, 15863}, {149, 214}, {354, 2771}, {474, 13205}, {484, 15325}, {497, 6951}, {499, 5445}, {517, 3582}, {528, 15015}, {551, 6175}, {942, 11571}, {1001, 3254}, {1145, 1698}, {1156, 5542}, {1210, 11009}, {1319, 3583}, {1385, 4857}, {1388, 9669}, {1420, 10483}, {1478, 9779}, {1479, 5731}, {1482, 12619}, {1537, 1768}, {1699, 2829}, {1702, 13913}, {1703, 13977}, {1706, 3035}, {2646, 13274}, {2800, 5603}, {2801, 11038}, {3036, 3632}, {3057, 6797}, {3065, 3649}, {3086, 5903}, {3244, 12531}, {3245, 3911}, {3303, 12331}, {3304, 12611}, {3336, 12515}, {3340, 12832}, {3485, 5083}, {3576, 5840}, {3584, 5919}, {3585, 12764}, {3622, 6224}, {3628, 13143}, {3646, 7162}, {3679, 5854}, {3825, 4861}, {3874, 12532}, {4297, 10724}, {4309, 13199}, {4316, 5126}, {4317, 12248}, {4330, 13624}, {4870, 5049}, {4973, 5180}, {4996, 5248}, {5131, 5298}, {5259, 13279}, {5425, 11019}, {5433, 11010}, {5550, 9802}, {5904, 10529}, {6595, 12267}, {7280, 12701}, {7284, 11372}, {7343, 13605}, {9964, 10122}, {10039, 10172}, {10074, 12047}, {10246, 11238}, {10265, 10698}, {10399, 12691}, {10404, 16128}, {10589, 12647}, {10595, 12247}, {10768, 11710}, {10769, 11711}, {10770, 11712}, {10771, 11713}, {10772, 11714}, {10773, 11716}, {10774, 11717}, {10775, 11718}, {10776, 11719}, {10777, 11700}, {10778, 11720}, {10779, 11721}, {10780, 11722}, {10912, 12641}, {11012, 16155}, {11256, 12607}, {11570, 14986}, {12767, 13226}, {13463, 13747}

= reflection of X(5131) in X(5298)

= incircle-inverse-of X(12019)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 11, 80), (1, 80, 7972), (1, 9897, 1317), (5, 12737, 12751), (11, 1317, 12019), (11, 1387, 1), (149, 3616, 214), (496, 12740, 10073), (946, 5563, 79), (1317, 12019, 9897), (1484, 5901, 6265), (6264, 8227, 119), (6326, 9624, 11729), (9897, 12019, 80), (11373, 11376, 1)

= [ 2.3255262506491480, 2.8119365333782980, 0.6206193815767199 ]

 

3)

A->Na = X(4)

Na->A = midpoint of X(11) and X(946)

= (b+c)*a^6+(b^2-6*b*c+c^2)*a^5- (b+c)*(4*b^2-9*b*c+4*c^2)*a^4- (2*b-c)*(b-2*c)*(b-c)^2*a^3+( b^2-c^2)*(b-c)*(5*b^2-3*b*c+5* c^2)*a^2+(b^2-c^2)^2*(b^2-3*b* c+c^2)*a-2*(b^2-c^2)^3*(b-c) : : (barys)

= 3*X(2)+X(14217), 3*X(11)+X(1537), 3*X(11)-X(10265), X(80)+3*X(5603), X(100)-5*X(8227), X(104)+3*X(1699), X(119)-3*X(3817), X(153)-9*X(9779), X(214)-3*X(5886), 3*X(381)+X(12737), 3*X(946)-X(1537), 3*X(946)+X(10265), X(1145)-3*X(10175), X(1538)+3*X(7743), 3*X(5886)+X(10738)

= on lines: {1, 6246}, {2, 14217}, {4, 11715}, {5, 2802}, {11, 65}, {80, 5603}, {100, 8227}, {104, 1699}, {119, 3817}, {153, 9779}, {214, 5886}, {381, 12737}, {496, 12005}, {515, 1387}, {516, 6713}, {517, 6702}, {952, 3850}, {1125, 5840}, {1145, 10175}, {1320, 5587}, {1482, 15863}, {1484, 2801}, {1512, 8068}, {2771, 13374}, {3036, 5087}, {3091, 12751}, {3576, 10724}, {3616, 12119}, {3898, 6980}, {5083, 5533}, {5541, 7988}, {6326, 10707}, {6667, 6684}, {6796, 9614}, {6918, 13205}, {7972, 10595}, {7989, 12653}, {7993, 10711}, {10057, 10598}, {10698, 11522}, {11236, 11256}, {11375, 13274}, {11376, 13273}, {12743, 15950}

= midpoint of X(i) and X(j) for these {i,j}: {1, 6246}, {4, 11715}, {11, 946}, {214, 10738}, {1482, 15863}, {1484, 12611}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (11, 1537, 10265), (946, 10265, 1537), (5533, 12047, 5083), (5886, 10738, 214)

= [ 0.3249898352552261, 0.4761613329123817, 3.1610190197733970 ]

 

César Lozada