HYACINTHOS 27170

[Antreas P. Hatzipolakis]:

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

Denote:

Bc = the reflection of B' in NC', 

Cb = the reflection of C' in NB'

 

Ca = the reflection of C' in NA', 

Ac = the reflection of A' in NC' 

 

Ab = the reflection of A' in NB', 

Ba = the reflection of B' in NA' 

N1, N2, N3 = the orthogonal projections of A, B, C on BcCb, CaAc, AbBa, resp.

 

N12, N13 = the orthogonal projections of N1 on AC, AB, resp
N23, N21 = the orthogonal projections of N2 on BA, BC, resp.
N31, N32 = thge ortogonal projections of N3 on CB, CA, resp.

L1, L2, L3 = the Euler lines of N1N12N13, N2N23N21, N3N31N32|

A*B*C* = the triangle bounded by L1, L2, L3

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

[César Lozada]:

1)
A->A* = X(54)

A*->A =
= (8*cos(2*A)-8*cos(4*A)-15)*cos(B-C)+6*cos(3*A)*cos(2*(B-C))-cos(3*(B-C))+10*cos(3*A) : : (trilinears)
= 5*S^4+(64*R^4-6*R^2*(SA+5*SW)+4*SA^2-7*SB*SC+SW^2)*S^2-(32*R^2*(R^2-SW)+7*SW^2)*SB*SC : : (barys)
= on line {523, 973}
= [ -1.0689951666789650, 2.9579258225124740, 2.0862520663275730 ]

2)
A'->A* =X(6153)

A*->A* =
= (34*cos(2*A)+3*cos(4*A)+5*cos(6*A)-21/2)*cos(B-C)+(-13*cos(A)+3*cos(3*A)-6*cos(5*A)+cos(7*A))*cos(2*(B-C))+(6*cos(2*A)+5*cos(4*A)-cos(6*A)-3/2)*cos(3*(B-C))-cos(3*A)*cos(4*(B-C))-cos(7*A)-19*cos(A)+6*cos(3*A)-10*cos(5*A) : : (trilinears)
= on line {973, 1510}
= [ 30.8260829696937000, 36.8719447234063200, -36.1134893895401500 ]


César Lozada