## Reflections and Parallels (Locus)

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• Let ABC be a triangle, P, P* two isogonal points, A B C the pedal triangle of P, and A*B*C* the cevian triangle of P*. Let A be the reflection of A* in A ,
Message 1 of 5 , Aug 28, 2005
Let ABC be a triangle, P, P* two isogonal points,
A'B'C' the pedal triangle of P, and A*B*C* the
cevian triangle of P*.

Let A" be the reflection of A* in A', and La the parallel
to AA* through A". Similarly La,Lc.

Which is the locus of P such that La,Lb,Lc are concurrent?
Locus of the point of concurrence?

Greetings

Antreas

--
• ... I think that the lines La,Lb,Lc are ALWAYS concurrent (ie the locus is the whole plane). Is it true??? Antreas --
Message 2 of 5 , Aug 29, 2005
[APH]:

>Let ABC be a triangle, P, P* two isogonal points,
>A'B'C' the pedal triangle of P, and A*B*C* the
>cevian triangle of P*.
>
>Let A" be the reflection of A* in A', and La the parallel
>to AA* through A". Similarly La,Lc.
>
>Which is the locus of P such that La,Lb,Lc are concurrent?
>Locus of the point of concurrence?

I think that the lines La,Lb,Lc are ALWAYS concurrent
(ie the locus is the whole plane).

Is it true???

Antreas
--
• Yes Antreas it is true and the lines are parallel if P lies on the circumcircle. It is more easy to prove first that the lines from A , B , C parallel to the
Message 3 of 5 , Aug 30, 2005
Yes Antreas it is true
and the lines are parallel if P lies on the circumcircle.

It is more easy to prove first that the lines
from A', B', C' parallel to the lines AA*,BB*,CC*
are ALWAYS concurrent and if Q is the point of
concurrence then your point of concurrence is the
reflection of P* in Q.

Best regards

> [APH]:
>
> >Let ABC be a triangle, P, P* two isogonal points,
> >A'B'C' the pedal triangle of P, and A*B*C* the
> >cevian triangle of P*.
> >
> >Let A" be the reflection of A* in A', and La the parallel
> >to AA* through A". Similarly La,Lc.
> >
> >Which is the locus of P such that La,Lb,Lc are concurrent?
> >Locus of the point of concurrence?
>
>
> I think that the lines La,Lb,Lc are ALWAYS concurrent
> (ie the locus is the whole plane).
>
> Is it true???
>
>
> Antreas
>
>
• ... Dear Nikos [welcome back from summer vacation!] I have parametrized it. See ANOPOLIS : http://groups.yahoo.com/group/Anopolis/ message #94 BTW, I added a
Message 4 of 5 , Aug 30, 2005
>Yes Antreas it is true
>and the lines are parallel if P lies on the circumcircle.
>
>It is more easy to prove first that the lines
>from A', B', C' parallel to the lines AA*,BB*,CC*
>are ALWAYS concurrent and if Q is the point of
>concurrence then your point of concurrence is the
>reflection of P* in Q.

Dear Nikos
[welcome back from summer vacation!]

I have parametrized it.

See ANOPOLIS : http://groups.yahoo.com/group/Anopolis/
message #94

BTW, I added a photograph in the group.
(Einai h tampela tou xwriou mou, diatrhth apo tis
mpalw8ies twn xwrianwn mou !!)

I think it is a good idea to add a photograph
in hyacinthos group as well. Maybe the photograph
of Emile Lemoine (or one of mine! :-))

Antreas

>Best regards
>
>> [APH]:
>>
>> >Let ABC be a triangle, P, P* two isogonal points,
>> >A'B'C' the pedal triangle of P, and A*B*C* the
>> >cevian triangle of P*.
>> >
>> >Let A" be the reflection of A* in A', and La the parallel
>> >to AA* through A". Similarly La,Lc.
>> >
>> >Which is the locus of P such that La,Lb,Lc are concurrent?
>> >Locus of the point of concurrence?
>>
>>
>> I think that the lines La,Lb,Lc are ALWAYS concurrent
>> (ie the locus is the whole plane).
>>
>> Is it true???
>>
>>
>> Antreas
>>
>>
>
>
>
>
>
>
>
>

--
• Dear Antreas & Nikolaos, ... [ND] ... [APH] ... Think that .. Locus of the points of concurrence is the line connecting P* and Q. The point of concurrence, pC,
Message 5 of 5 , Aug 30, 2005
Dear Antreas & Nikolaos,

[APH]:
>Let ABC be a triangle, P, P* two isogonal points,
>A'B'C' the pedal triangle of P, and A*B*C* the
>cevian triangle of P*.
>
>Let A" be the reflection of A* in A', and La the parallel
>to AA* through A". Similarly La,Lc.
>
>Which is the locus of P such that La,Lb,Lc are concurrent?
>Locus of the point of concurrence?
>
> I think that the lines La,Lb,Lc are ALWAYS concurrent
> (ie the locus is the whole plane).
>
> Is it true???

[ND]
>Yes Antreas it is true
>and the lines are parallel if P lies on the circumcircle.
>
>It is more easy to prove first that the lines
>from A', B', C' parallel to the lines AA*,BB*,CC*
>are ALWAYS concurrent and if Q is the point of
>concurrence then your point of concurrence is the
>reflection of P* in Q.

[APH]
>I have parametrized it.
>
>Let ABC be a triangle, P, P* two isogonal points,
>A'B'C' the pedal triangle of P, and A*B*C* the
>cevian triangle of P*.

>Let A", B", C" be points on BC, CA, AB resp. such that

>A"A' / A"A* = B"B' / B"B* = C"C' / C"C* = t,

>and La,Lb,Lc parallels to AA*,BB*,CC* through A",B",C", resp.

>Are the lines La,Lb,Lc concurrent for all t's ?

>Locus of point of concurrence as t varies?

Think that ..
Locus of the points of concurrence is the line connecting P* and Q.

The point of concurrence, pC, is such that |pCQ| / |pCP*| = t

Best regards,
Peter.
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