HYACINTHOS 25129

[Antreas P. Hatzipolakis]:

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

Denote

Aa, Bb, Cc = the reflections of A, B, C in B'C', C'A', A'B', resp.

Aaa, Bbb, Ccc = the reflections of Aa,Bb,Cc in BC, CA, AB, resp.

A1, B2, C3 = the reflections of A,B,C in BC, CA, AB, resp.

A11, B22, C33 = the reflections of A1, B2, C3 in B'C', C'A', A'B', resp.

A*B*C* = the triangle bounded by AaaA11, BbbB22, CccC33

1. ABC, AaaBbbCcc are perspective.

Perspector?

 

2. ABC, A11B22C33 are perspective.

Perspector (on the OI line)?

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

The prallelogic center (ABC, A*B*C*) is the I

The other one?

 

[César Lozada]:

1)      X(79)

2)      X(35)

3)      X(1) and

(A*->A) = (p^2+q*p+1/4)*(4*q*p^5-p^4-(4* q^2+5)*q*p^3+2*q^2*p^2+(4*q^2+ 1)*q*p-3/2*q^2+9/16) : : (trilinears), where p=sin(A/2), q=cos((B-C)/2)

= [ 0.349324465921745, 0.87629098804711, 2.872774813603269 ]

 

César Lozada