- 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

-- - [APH]:

>Let ABC be a triangle, P, P* two isogonal points,

I think that the lines La,Lb,Lc are ALWAYS concurrent

>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?

(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 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

Nikolaos Dergiades

> [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

Dear Nikos

>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.

[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

--

>Nikolaos Dergiades

>

>> [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

>>

>>

>

>

>

>

>Yahoo! Groups Links

>

>

>

>

- Dear Antreas & Nikolaos,

[APH]:>Let ABC be a triangle, P, P* two isogonal points,

[ND]

>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???

>Yes Antreas it is true

[APH]

>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.

>I have parametrized it.

Think that ..

>

>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?

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.