HYACINTHOS 20331

Dear All My Friends,
 
Reflect point P in BC, CA, AB of reference triangle ABC we
get reflection triangle PaPbPc. Q is isogonal conjugate of P wrt triangle
PaPbPc.
 
Denote the transform as Q = gR(P).
 
If barycentrics of P = {p : q : r} then barycentrics of Q:
 
a^2*(p*(b^2*c^2*p - c^2*q*(c^2 - a^2) - b^2*r*(b^2 - a^2)) +
q*r*(a^2*(b^2 + c^2) - (b^2 - c^2)^2))  :  :
 
Some special cases
 
P = X(1)
Barycentrics of Q:
a*((a – b + c)*(a + b - c)*(b + c) – a*b*c)  :  :
Search value: -0.6308558824469 not in current ETC.
 
P = X(2)
Barycentrics of Q:
a^2*(2*a^2*(b^2 + c^2) - 2*(b^2 - c^2)^2 - b^2*c^2)  :  :
Search value: -0.715622153886 not in current ETC.
 
P = X(3), Q = X(4)
P = X(4), Q = X(4)
P = X(5), Q = X(143)
 
P = X(6)
Barycentrics of Q:
3*a^4 - (b^2 - c^2)^2  :  :
Search value: -0.3718585039795 not in current ETC.
 
Any other interesting facts of the transform gR(P)?
 
Best regards,
Bui Quang Tuan