## Reflections in the sidelines of the circumcevian triangle

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• For the paper on the 20th Anniversary of Hyacinthos....... :-) Let ABC be a triangle, P,P* two isogonal conjugate points and A B C the circumcevian triangle
Message 1 of 2 , Mar 7, 2013
For the paper on the 20th Anniversary of Hyacinthos....... :-)

Let ABC be a triangle, P,P* two isogonal conjugate points
and A'B'C' the circumcevian triangle of P.

Which is the locus of P such that the reflections of

1. AP, BP, CP

2. AP*, BP*, CP*

in the respective sidelines B'C', C'A', A'B' of A'B'C' are concurrent?

APH
• This time I include the Mathematica instructions. We can find the loci: 1. circumcircle + nonic 2. circuncircle + McCay cubic ... I think that the code is neat
Message 2 of 2 , Mar 7, 2013
This time I include the Mathematica instructions. We can find the loci:

1. circumcircle + nonic
2. circuncircle + McCay cubic

-----

I think that the code is neat enough:

Det - Determinant of a matrix (included in Mathematica)

ConjugadoIsogonal - Isogonal conjugate of a point
TrianguloCircunceviano - Circuncevian triangle of a point
Recta - Line through two points
SimetriaAxial - Reflection of a point on the line through two points

----------

<< Baricentricas`;

ptP = {x, y, z};
{ptA1, ptB1, ptC1} = TrianguloCircunceviano[ptP];

Factor[Det[{
Recta[SimetriaAxial[ptA, ptB1, ptC1],
SimetriaAxial[ptP, ptB1, ptC1]],
Recta[SimetriaAxial[ptB, ptC1, ptA1],
SimetriaAxial[ptP, ptC1, ptA1]],
Recta[SimetriaAxial[ptC, ptA1, ptB1],
SimetriaAxial[ptP, ptA1, ptB1]]}]]

Factor[Det[{
Recta[SimetriaAxial[ptA, ptB1, ptC1],
SimetriaAxial[ptQ, ptB1, ptC1]],
Recta[SimetriaAxial[ptB, ptC1, ptA1],
SimetriaAxial[ptQ, ptC1, ptA1]],
Recta[SimetriaAxial[ptC, ptA1, ptB1],
SimetriaAxial[ptQ, ptA1, ptB1]]}]]
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