HYACINTHOS 26302

 

[Antreas P. Htzipolakis]:

 

Orthic version:

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

Denote:

Na = the NPC center of AB'C'. Similarly Nb, Nc
N1 = the reflection of Na in AA'. Similarly N2, N3
La = the Euler line of AB'C'. Similarly Lb, Lc
L1 = the reflection of La in AA'. Similarly L2, L3
.
A*B*C* = the triangle bounded by L1, L2, L3

 

1. The centroid of N1N2N3 lies on the circumcenter-incenter line of A'B'C' = NH line (Euler line) of ABC.

2. A'B'C', A*B*C* are parallelogic.

 

Excentral version:

 

Let ABC be a triangle and  IaIbIc the antipedal triangles of I.

Denote:

Na = the NPC center of IaBC. Similarly Nb, Nc
N1 = the reflection of Na in AIa. Similarly N2, N3
La = the Euler line of IaBC. Similarly Lb, Lc
L1 = the reflection of La in AIa. Similarly L2, L3
.
A*B*C* = the triangle bounded by L1, L2, L3

1. The centroid of N1N2N3 lies on the OI line of ABC.

2. ABC, A*B*C* are parallelogic.

[César Lozada]:

 

 

Orthic version:

 

1)      Centroid of N1N2N3 = X(13490)

2)      A'B'C', A*B*C* are parallelogic.

 

Centers:

P(A’->A*) = X(403)

 

P(A*->A’) = X(4)X(51) ∩ X(30)X(125)

= (-a^2+b^2+c^2)*(2*a^8-(b^2+c^ 2)*a^6-2*(b^2-c^2)^2*a^4-(b^4- c^4)*(b^2-c^2)*a^2+2*(b^2-c^2) ^4) : : (barycentrics)

= cos(A)*(3*cos(A)*cos(B-C)-cos( 2*(B-C))-cos(2*A)-1) : : (trilinears)

= 2*X(265)+X(1531) = X(3292)-4*X(10297) = X(13399)+2*X(13473)

= On lines: {4,51}, {5,13367}, {30,125}, {115,8779}, {184,381}, {265,1531}, {382,1204}, {403,1495}, {511,3153}, {546,6146}, {578,7547}, {1092,12293}, {1181,3843}, {1425,3585}, {1503,10151}, {1568,3292}, {1594,13403}, {1650,12096}, {2071,10733}, {3270,3583}, {3410,5907}, {3574,12241}, {3818,6467}, {3830,10605}, {3839,5476}, {3850,8254}, {5562,9927}, {5622,11645}, {7507,11424}, {7577,11430}, {10255,12038}, {10282,12289}, {11017,11577}, {12022,13366}, {12828,13202}

= midpoint of X(i) and X(j) for these {i,j}: {2071,10733}, {13202,13399}

= reflection of X(i) in X(j) for these (i,j): (403,7687), (1495,403), (1568,10297), (3292,1568), (13202,13473)

= [ -1.099897882200544, -1.49832221196074, 5.185609651203605 ]

 

Excentral version:

 

1)      Centroid of N1N2N3 = X(10202)

2)      ABC, A*B*C* are parallelogic.  Centers:  P(A->A*) = X(36)  &  P(A*->A) = X(4511)

 

César Lozada