## Re: [EMHL] Locus defined by trilinears' equation

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• Dear Nikolaos Yes ; In others words, two distinct points have the same tripolar coordinates if they are inverse wrt the circumcircle. So the restriction of f
Message 1 of 38 , Mar 24, 2005
Dear Nikolaos
Yes ;
In others words, two distinct points have the same tripolar
coordinates if they are inverse wrt the circumcircle.

So the restriction of f or g to the circumcircle and its interior is
one to one onto the range which is compact since f or g is continuous.

I just give this riddle so that every one thinks about Ptolemy's
Theorem as Ptolemy Inequality Theorem and not as Ptolemy Equality
theorem which is only a cocyclicity
condition.
I just have this idea when I read Zbaruh's question.
Best regards and nice Spring for all Hyacinthos friends
especially those who live in Tokyo marveling at cherries blossoming!
François
• Dear friends I notice the following fact certainly well known for a long time and very easy to prove: Call P any point in the plane of triangle ABC and L the
Message 38 of 38 , Apr 22, 2005
Dear friends
I notice the following fact certainly well known for a long time and very
easy to prove:

Call P any point in the plane of triangle ABC and L the trilinear polar of P
wrt ABC.
Call Gamma the inscribed conic in ABC of which P is the perspector or
Brianchon point, i.e : P is the perspector of ABC and the contact triangle.
Let M be any point on Gamma and call T the tangent in M at Gamma.
Then the trilinear pole O of T is on L.
From this fact can we find :
1°a simple projective construction of the tangent T of M at Gamma.
2°If O is any point on L, a simple projective construction of the contact
point M of the trilinear polar T of O wrt ABC with Gamma.