- Dear Peter

>> [APH]:

[PM]:

>> >> Let ABC be a triangle and P a point.

>> >>

>> >> Which is the locus of P such that the reflections of:

>> >> PA in BC, PB in CA, and PC in AB, are concurrent?

>>

>> Now, let A'B'C' be the medial triangle of ABC.

>>

>> Which is the locus of P such that the reflections of:

>> PA in B'C', PB in C'A', and PC in A'B', are concurrent?

>>

>> It is well known that the line at infinity is part of the

>> locus, since three parallels through A,B,C, reflected on

>> the sidelines of the medial triangle concur at

>> the Nine Point Circle of ABC.

>Another part is the Jerabek hyperbola with perpector on the Euler

We can replace the medial triangle with the orthic, and ask for

>line.

the locus etc

In general:

Let ABC be a triangle, Q = (u:v:w) a fixed point, QaQbQc its pedal

triangle, and P a variable point.

Which is the locus of P such that the reflections of:

PA in QbQc, PB in QcQa, and PC in QaQb, are concurrent?

Antreas

-- - Dear Antreas,

[APH]>

Orthic, 2 cubics one, of which passes through X{3,24,186,1299,2931}.

> We can replace the medial triangle with the orthic, and ask for

> the locus etc

>

> In general:

>

> Let ABC be a triangle, Q = (u:v:w) a fixed point, QaQbQc its pedal

> triangle, and P a variable point.

>

> Which is the locus of P such that the reflections of:

> PA in QbQc, PB in QcQa, and PC in QaQb, are concurrent?

Anticomp, 2 cubics, one of which is Bernards K025.

Excentral, whole plane, perspector the isogonal of P.

General case looks as if it could be messy.

Best regards,

Peter. - Dear Antreas
> In general:

A circular circumcubic going through the isogonal conjugate of Q.

>

> Let ABC be a triangle, Q = (u:v:w) a fixed point, QaQbQc its pedal

> triangle, and P a variable point.

>

> Which is the locus of P such that the reflections of:

> PA in QbQc, PB in QcQa, and PC in QaQb, are concurrent?

Jean-Pierre