HYACINTHOS 26090

[Antreas P. Hatzipolakis]:

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

Let A", B", C" be three arbitrary points on AA', BB', CC'.

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

Obviously ABC, NaNbNc are orthologic with orthologic center (NaNbNc, ABC) = N.


Applications:

1. Let A", B", C" be the midpoints of AA', BB', CC', resp.
(A"B"C" = the midheight triangle)

2. Let A" be the intersection of B'C' and AA' and similarly B", C"
(A"B"C" = the cevian triangle of H wrt triangle A'B'C')

 

3. Let A", B", C" be the midpoints of AH, BH, CH, resp.
(A"B"C" = the Euler triangle of ABC)

Which are the orthologic centers (ABC, NaNbNc) ?

[Peter Moses]:

 

Hi Antreas,

 

1).

X(64).

 

2 = 3).

b^2 c^2 (-a^2+b^2-c^2)^2 (a^2+b^2-c^2)^2 (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4):: 

on lines {4,51},{5,324},{54,436},{93, 1487},{107,1141},{235,2970},{ 264,3090},{327,1235},{393,847} ,{467,8800},{1173,4994},{1594, 3613},{1629,11816},...}.

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1075,5890),(4,3168,3567),( 51,8887,4),(107,8884,3518),( 1093,2052,4),... .

on ABCHN.

polar conjugate of X(97).

X(2052)-Ceva conjugate of X(53).

X(i)-cross conjugate of X(j) for these (i,j): {{53, 324}, {6750, 4}}.

isoconjugate of X(j) and X(j) for these (i,j): {{3, 2169}, {48, 97}, {54, 255}, {275, 4100}, {394, 2148}, {577, 2167}, {1092, 2190}, {6507, 8882}}.

barycentric product X(i)X(j) for these {i,j}: {{4, 324}, {5, 2052}, {53, 264}, {311, 393}, {343, 1093}, {467, 847}, {823, 2618}, {1969, 2181}, {6528, 12077}}.

barycentric quotient X(i)/X(j) for these {i,j}: {{4, 97}, {5, 394}, {19, 2169}, {51, 577}, {53, 3}, {158, 2167}, {216, 1092}, {311, 3926}, {324, 69}, {343, 3964}, {393, 54}, {467, 9723}, {1093, 275}, {1096, 2148}, {1393, 7125}, {1953, 255}, {2052, 95}, {2181, 48}, {3199, 184}, {6520, 2190}, {6524, 8882}, {6529, 933}, {7069, 2289}, {12077, 520}}.

 

Best regards,

Peter Moses.