Re: [EMHL] Fuss Theorem for Bicentric Quadrilateral
- View SourceDear Prof. Nikolaos:
Thank you very much for answer me.
Perhaps I did not know how to explain the reason of my question.
The method that I developed does not use trigonometry. I like resolving with
the traditional methods of the euclidean geometry
The method that mentions the Prof. Paul Yiu in his e-book Euclidean
Geometry pag.159-161 also utilizes trigonometry.
Do I can use the inversion method?
Do you know another method without use trigonometry?
I will thank your answer
Juan Carlos Salazar
----- Original Message -----
From: "Nikolaos Dergiades" <ndergiades@...>
Sent: Tuesday, March 18, 2003 5:52 PM
Subject: Re: [EMHL] Fuss Theorem for Bicentric Quadrilateral
> Dear Juan,
> I don't know the who you are asking but
> I think that it is a simple ralation.
> Because since A + C = 180
> then sin(C/2) = cos(A/2) and hence
> 1/AI² + 1/CI² = sin²(A/2)/r² + sin²(C/2)/r²
> = [sin²(A/2) + cos²(A/2)]/r² = 1/r²
> Similarly for the sum 1/BI² + 1/DI² = 1/r²
> Best regards
> Nikolaos Dergiades
> >Dear friends:
> >I am new member.
> >I did read the Fuss Theorem to bicentric quadrilateral in
> the book
> >Introduction to Modern Geometry by Levi Shively in the year
> >without proof.
> >I have developed in the same year this relation:
> >(I: incenter and r: inradii,in ABCD bicentric
> quadrilateral) and
> >then I did develop the proof to Fuss theorem with
> utilization of
> >power of points and Appolonio's theorem.
> >I don't know one paper on this relation is writed.
> >I thing this condition "necesary and sufficent" for the
> >quadrilateral be a bicentric quadrilateral, when the sides
> >tangents to one circle.
> >Do you send me, Who develop by first time this above
> >Juan Carlos Salazar
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