HYACINTHOS 25986

[Antreas P. Hatzipolakis]:

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

Denote:

Ab, Ac = the orthogonal projections of A' on BB', CC', resp.

(Nab), (Nac) = the NPCs of AbB'C', AcB'C', resp.

Ra = the radical axis of (Nab), (Nac). Similarly Rb, Rc.

A*B*C* = the triangle bounded by Ra, Rb, Rc.

ABC, A*B*C* are parallelogic.
Parallellogic centers?


[Angel Montesdeoca]:

*** Parallellogic center of ABC  with respect to A*B*C* is X(12)

*** Parallellogic center of A*B*C*  with respect to ABC is 

V = (r+2R) X(1) + 3r X(2)

V = ( 2 a^4-a^3 (b+c)-a^2 (3 b^2+2 b c+3 c^2)+a (b-c)^2 (b+c)+(b^2-c^2)^2: ... : ...),

 

V is the  midpoint of X(i) and X(j) for these {i,j}:  {1,10039}, {12,2646}, {35,12047}, {4870,4995}
V lies on lines: {1,2}, {3,226},{ 4,3601,4304}, {5,950}, {7,3523}, {9,6857}, {12,515}, {20,5226}, {21,908}, {29,1785}, {35,411}, {36,4298}, {37,216}, {40,3485}, {46,3671}, {55,946}, {56,10165}, {57,631}, {58,3074}, {63,6910}, {65,5432}, {72,5745}, {84,6935}, {86,1167}, {140,942}, {142,474}, {165,4295}, {255,307}, {354,5433}, {376,5714}, {377,4855}, {388,3576}, {390,9614}, {404,5249}, {405,3452}, {442,5440}, {443,5438},{ 452,5748}, {495,1385}, {496,11230}, {497,6864}, {518,4999}, {527,3916}, {549,553}, .....

 

 

with (6,9,12)-search numbers (1.91083147220420, 1.70501004030684, 1.57835069760078).

Angel Montesdeoca

 

[Peter Moses]:¨

 

Hi Antreas,

 

A few extras ....

 

V = 2 a^4-a^3 b-3 a^2 b^2+a b^3+b^4-a^3 c-2 a^2 b c-a b^2 c-3 a^2 c^2-a b c^2-2 b^2 c^2+a c^3+c^4::

Complement X(6734).

on lines {{1,2},{3,226},{4,3601},{5, 950},{7,3523},{9,6857},{12, 515},{20,5226},{21,908},{29, 1785},{35,411},{36,4298},{37, 216},{40,3485},{46,3671},{55, 946},{56,10165},{57,631},{58, 3074},{63,6910},{65,5432},{72, 5745},{84,6935},{86,1167},{ 140,942},{142,474},{165,4295}, {255,307},{354,5433},{376, 5714},{377,4855},{388,3576},{ 390,9614},{404,5249},{405, 3452},{442,5440},{443,5438},{ 452,5748},{495,1385},{496, 11230},{497,6864},{518,4999},{ 527,3916},{549,553},{750,1771} ,{940,7078},{943,6905},{944, 6956},{960,6690},{962,5281},{ 965,5257},{991,1745},{993, 12527},{1012,6260},{1056,1420} ,{1058,10389},{1071,6705},{ 1150,4101},{1155,3649},{1323, 1446},{1335,8983},{1445,3338}, {1478,3612},{1479,3817},{1490, 6847},{1656,5722},{1697,5603}, {1699,4294},{1728,3305},{1735, 2292},{1770,5010},{1788,11529} ,{1836,5217},{1837,10175},{ 1838,7513},{1892,3515},{2099, 11362},{3035,3812},{3075,7572} ,{3090,3488},{3091,3586},{ 3158,5082},{3295,5886},{3303, 11376},{3306,6921},{3333,3475} ,{3336,11551},{3340,5657},{ 3419,12437},{3486,5587},{3524, 4654},{3525,11518},{3530,3982} ,{3583,6894},{3585,6895},{ 3614,10543},{3628,12433},{ 3686,5742},{3742,6691},{3746, 5443},{3772,4255},{3822,10523} ,{3940,5791},{4293,5290},{ 4301,5119},{4305,5691},{4652, 5905},{4870,4995},{4909,5740}, {5044,6675},{5054,5708},{5083, 6713},{5084,5436},{5129,5328}, {5135,9028},{5204,10404},{ 5248,8069},{5252,5882},{5261, 5731},{5265,11037},{5294, 11031},{5316,11108},{5425, 5445},{5435,10303},{5439,6692} ,{5444,5563},{5482,11573},{ 5570,6681},{5709,5761},{5717, 5718},{5720,6824},{5727,5818}, {5728,6666},{5730,5837},{5732, 8232},{5735,5766},{5747,8804}, {5750,5830},{5777,10391},{ 5850,6763},{5901,9957},{5902, 12563},{6198,7537},{6245,6833} ,{6282,6908},{6710,11028},{ 6718,12016},{6767,11373},{ 6828,7951},{6834,7682},{6837, 7675},{6846,10382},{6860, 10827},{6861,10395},{6878, 11048},{6890,10884},{6892, 7330},{6926,8726},{6966,8545}, {6991,7741},{7308,10396},{ 7498,7952},{7742,12573},{7962, 10595},{7988,10591},{10086, 11599},{10265,12739},{10399, 11020},{10956,11715},{11507, 12609}}.

Midpoint of X(i) and X(j) for these {i,j}: {{1, 10039}, {12, 2646}, {35, 12047}, {4870, 4995}}.

X[1] + 3 X[3584], 5 X[3616] - X[4861], 3 X[3584] - X[10039].

(r + 2 R) X[1] + 3 r X[2], r X[3] + (r + R) X[226], r X[4] + (3 r + 2 R) X[3601], 2 r X[5] + R X[950], (r + 4 R) X[7] + 7 r X[3523], (r+4 R) X[9] - (5 r + 6 R) X[6857], r X[20] + (7 r+4 R) X[5226].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,1210),(1,498,10),(1,499, 11019),(1,1737,6738),(1,3584, 10039),(1,3624,3086),(1,12647, 3244),(2,8,5705),(2,78,10),(2, 5703,1),(3,226,4292),(3,11374, 226),(4,3601,4304),(20,5226, 9612),(21,908,12572),(55,946, 10624),(55,11375,946),(65, 5432,6684),(72,7483,5745),( 140,942,3911),(140,5719,942),( 307,5736,3664),(376,5714,9579) ,(388,3576,4311),(404,5249, 12436),(495,1385,10106),(631, 3487,57),(944,8164,9578),( 1125,6700,2),(1478,3612,4297), (1770,5010,12512),(3035,11281, 3812),(3090,3488,9581),(3091, 4313,3586),(3295,5886,12053),( 3475,7288,3333),(3485,5218,40) ,(3486,10588,5587),(3601,5219, 4),(3634,6738,1737),(3671, 10164,46),(3817,4314,1479),( 3947,4297,1478),(4305,10590, 5691),(5261,5731,9613),(5290, 7987,4293),(5761,6954,5709),( 10303,11036,5435).

X(i)-complementary conjugate of X(j) for these (i,j): {{943, 1329}, {1175, 960}, {2259, 3452}, {2982, 141}}.

 

Best regards,

Peter Moses.