HYACINTHOS 25968

[Antreas P. Hatzipolakis]:

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

Denote:

Ab, Ac = the orthogonal projections of B', C' on AA', resp.
Bc, Ba = the orthogonal projections of C', A' on BB', resp.
Ca, Cb = the orthogonal projections of A', B' on CC', resp.

A*B*C* = the triangle bounded by BcCb, CaAc, AbBa

A**B**C** = the triangle bounded by BaCa, CbAb, AcBc

ABC, A*B*C* are bilogic (perspective and orthologic)
Perspector, orthologic centers ?

ABC, A**B**C** are homothetic.
Homothetic center?

Orthologic center (A**B**C**, ABC) = orthocenter of A**B**C** ?

 

[Peter Moses]:

 

 

Hi Antreas,

 

>ABC, A*B*C* are bilogic (perspective and orthologic)

>Perspector

X(2346)

>orthologic centers ?
X(79) &

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

on lines {{1,4},{2,3338},{3,10404},{5, 354},{7,46},{8,12559},{10, 3681},{11,5045},{12,942},{35, 4292},{36,4298},{40,4654},{55, 1770},{56,6883},{57,498},{58, 3011},{63,10198},{65,495},{79, 516},{80,6738},{142,1698},{ 191,527},{210,8728},{355, 11237},{377,3811},{442,518},{ 484,3982},{499,3333},{517, 3649},{519,5178},{529,11281},{ 551,3897},{553,3336},{726, 3178},{908,1125},{938,10590},{ 999,11375},{1071,7680},{1089, 3912},{1103,4328},{1145,10107} ,{1210,3947},{1329,5439},{ 1330,3757},{1385,5434},{1714, 3751},{1718,5262},{1738,3293}, {1788,8164},{1836,3295},{1837, 9654},{1892,11398},{2476,3873} ,{2646,5719},{2801,10122},{ 2886,3555},{3086,5226},{3090, 3296},{3091,11038},{3303, 12699},{3304,5886},{3337,3911} ,{3340,12647},{3452,3624},{ 3576,4317},{3579,11246},{3600, 6992},{3601,4299},{3612,4293}, {3634,5557},{3636,11813},{ 3670,5530},{3671,5903},{3697, 3826},{3742,4187},{3753,12607} ,{3754,6735},{3782,3931},{ 3822,3874},{3824,3925},{3889, 11680},{3916,6690},{3936,4968} ,{4004,8256},{4294,10578},{ 4295,5119},{4302,9579},{4304, 10483},{4309,10389},{4338, 6361},{4415,6051},{4870,5901}, {5049,9955},{5083,8068},{5252, 12645},{5259,12572},{5261, 6993},{5425,12563},{5443, 12577},{5586,9588},{5587, 11518},{5687,5880},{5722, 10895},{5745,6763},{5905, 12514},{6745,12436},{6767, 12701},{6825,12704},{6831, 12675},{6847,10085},{6890, 7284},{7373,11376},{7741, 11019},{7958,10157},{8227, 10072},{8727,12680},{9578, 10573},{10052,10075},{10580, 10591},{10587,11415},{11010, 11552},{12528,12617}}.

midpoint of X(i) and X(j) for these {i,j}: {{1, 5270}, {79, 3746}}.

X[5178] - 3 X[6175].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,226,12047),(1,1478,10572),( 1,3585,950),(1,5290,1478),(1, 9612,1479),(4,3475,1),(7,3085, 46),(12,942,1737),(65,495, 10039),(65,6147,11551),(388, 3487,1),(388,10629,1478),(495, 6147,65),(553,6684,3336),(938, 10590,10826),(1056,3485,1),( 1125,12527,5251),(1210,3947, 7951),(2476,3873,10916),(3333, 5219,499),(3336,3584,6684),( 3681,4197,10),(3822,3874,6734) ,(3947,5542,1210),(4293,5703, 3612),(5226,11037,3086),(9578, 11529,10573),(10039,11551,65).

crosspoint of X(92) and X(1268).

crosssum of X(48) and X(2308).

 

>ABC, A**B**C** are homothetic.

X(81).

>Orthologic center (A**B**C**, ABC) = orthocenter of A**B**C** ?

2 a^7-3 a^5 b^2-a^4 b^3+2 a^2 b^5+a b^6-b^7-3 a^4 b^2 c-2 a^3 b^3 c+2 a^2 b^4 c+2 a b^5 c+b^6 c-3 a^5 c^2-3 a^4 b c^2-4 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2-a^4 c^3-2 a^3 b c^3-4 a^2 b^2 c^3-4 a b^3 c^3-3 b^4 c^3+2 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a^2 c^5+2 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7:: 

on lines {{1,30},{3,5713},{4,81},{5, 1724},{58,6841},{155,5706},{ 186,229},{225,6357},{283,442}, {323,2475},{355,3564},{381, 5292},{382,5733},{407,1437},{ 540,10916},{580,6881},{582, 8728},{942,1835},{1478,3157},{ 2003,3585},{3109,11657},{3332, 6850},{4340,6851},{5230,9958}, {5712,6869},{9840,11249}}.

 

Best regards,

Peter Moses.