## [EMHL] Re: Parry Locus

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• Dear Peter ... 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,
Message 1 of 7 , Sep 8 6:26 AM
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Dear Peter

>> [APH]:
>> >> 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.

[PM]:
>Another part is the Jerabek hyperbola with perpector on the Euler
>line.

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?

Antreas

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• Dear Antreas, [APH] ... Orthic, 2 cubics one, of which passes through X{3,24,186,1299,2931}. Anticomp, 2 cubics, one of which is Bernards K025. Excentral,
Message 2 of 7 , Sep 8 7:21 AM
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Dear Antreas,

[APH]
>
> 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?

Orthic, 2 cubics one, of which passes through X{3,24,186,1299,2931}.
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 ... A circular circumcubic going through the isogonal conjugate of Q. Jean-Pierre
Message 3 of 7 , Sep 8 8:25 AM
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Dear Antreas
> 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?

A circular circumcubic going through the isogonal conjugate of Q.
Jean-Pierre
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