HYACINTHOS 27171

 

[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.

La, Lb, Lc = the Euler lines of AN12N13, BN23N21, CN31N32|

A*B*C* = the triangle bounded by La, Lb, Lc

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

 

[César Lozada ]:

1) 

A->A* = X(74)

 

A*->A = 

= (72*cos(2*A)+8*cos(4*A)+65)* cos(B-C)+(-16*cos(A)-10*cos(3* A))*cos(2*(B-C))-cos(3*(B-C))- 22*cos(3*A)-96*cos(A) :: (trilinears)

= 11*S^4+(192*R^4-2*R^2*(29* SA+17*SW)+12*SA^2-9*SB*SC-SW^ 2)*S^2-3*(32*R^2*(3*R^2-SW)+3* SW^2)*SB*SC : : (barys

= on line {523, 974}

= [ 7.3808467095318320, 8.0603207640272280, -5.3461022206645470 ]

 

2)

A'->A* = X(11557)

 

A*->A' = 

= (164*cos(2*A)+59*cos(4*A)+5* cos(6*A)+217/2)*cos(B-C)-(103* cos(A)+43*cos(3*A)+8*cos(5*A)- cos(7*A))*cos(2*(B-C))+(12* cos(2*A)+5*cos(4*A)-cos(6*A)+ 15/2)*cos(3*(B-C))-cos(3*A)* cos(4*(B-C))-cos(7*A)-113*cos( A)-76*cos(3*A)-16*cos(5*A) : : (trilinears)

= on lines: {974, 1510}

= [ 6.9127789949475100, 6.8855859179531910, -4.3167929974205240 ]

 

César Lozada