HYACINTHOS 27090

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of I

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

MaMbMc = the midway triangle of I.
(ie Ma, Mb, Mc = the midpoints of AI, BI, CI, resp.)

N1, N2, N3 = the midpoints of NaMa, NbMb, NcMc, resp.

A'B'C', N1N2N3 are homothetic.
Homothetic center?

[César Lozada]:

 

 

Q = X(1)X(5) ∩ X(7)X(21)

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

= (R+2*r)*X(1)+2*r*X(5)

= on lines: {1, 5}, {2, 2099}, {7, 21}, {8, 7504}, {10, 11011}, {34, 1904}, {36, 7508}, {55, 5603}, {57, 5298}, {65, 392}, {140, 5903}, {145, 10588}, {214, 11112}, {226, 535}, {388, 1388}, {396, 7052}, {404, 14882}, {484, 549}, {497, 6839}, {517, 5432}, {547, 11545}, {632, 5445}, {944, 10895}, {946, 2646}, {953, 5397}, {962, 5217}, {997, 3925}, {999, 7489}, {1000, 1389}, {1056, 6965}, {1058, 6900}, {1155, 10165}, {1210, 14563}, {1357, 11731}, {1358, 11730}, {1359, 11733}, {1361, 11734}, {1362, 11726}, {1364, 11727}, {1385, 7354}, {1420, 10404}, {1454, 5250}, {1457, 3720}, {1468, 7277}, {1478, 10246}, {1479, 10543}, {1621, 5172}, {1656, 10573}, {1699, 13384}, {1737, 11230}, {1770, 13624}, {1788, 4323}, {1836, 3576}, {2320, 11114}, {2886, 4511}, {3022, 11728}, {3023, 11724}, {3027, 11725}, {3028, 11735}, {3057, 13411}, {3086, 6852}, {3303, 5703}, {3304, 3487}, {3324, 11732}, {3340, 3624}, {3476, 5226}, {3486, 7548}, {3488, 11238}, {3582, 5425}, {3601, 5805}, {3612, 12699}, {3636, 10106}, {3656, 4995}, {3671, 4031}, {3698, 6700}, {3754, 13747}, {3869, 4999}, {3871, 9802}, {3877, 6690}, {3919, 6681}, {4295, 5204}, {4305, 12953}, {4313, 9670}, {4654, 13462}, {4861, 12607}, {4975, 6358}, {5123, 10222}, {5183, 10164}, {5221, 7288}, {5561, 10483}, {5563, 6147}, {5690, 11009}, {5714, 9657}, {5731, 12943}, {5902, 15325}, {5905, 11194}, {5919, 13405}, {6357, 10571}, {6884, 14986}, {7967, 10590}, {8162, 10578}, {8581, 10177}, {8703, 15228}, {8715, 12732}, {9615, 9649}, {9955, 10572}, {10072, 15934}, {10199, 12832}, {10247, 12647}, {10527, 12635}, {10528, 10912}, {10532, 10953}, {10596, 10947}

= midpoint of X(1) and X(7951)

= X(7951) of anti-Aquila triangle

= X(11454) of Hutson intouch triangle

= X(11464) of intouch triangle

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 5, 10950), (1, 12, 10944), (1, 5219, 5252), (1, 5252, 1317), (1, 5443, 5), (1, 5886, 11), (1, 7988, 5727), (1, 8227, 1837), (1, 9624, 11376), (1, 11374, 15888), (1, 11375, 12), (495, 10283, 1), (1387, 5719, 1), (1837, 8227, 7173), (5252, 11375, 5219), (5719, 5901, 1387)

= [ 1.4190866274130090, 1.2114017600480280, 2.1470386661451260 ]

 

N1N2N3 and the  following triangles are parallelogic at the given centers: (Fuhrmann, 11729, 4), (1st Parry, 5901, 9810), (2nd Parry, 5901, 9811), (2nd Sharygin, 1125, 2254)

 

César Lozada