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• Dear Jean-Pierre and Mathlinkers, working on a generalization of a Nagel s result, I found in the archive of Hyacinthos the message # 19640 Dear Chris [JP]
Feb 24 1 of 2
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working on a generalization of a Nagel's result, I found in the archive of Hyacinthos the message # 19640
Dear Chris
[JP]
> > Consider a quadrilateral A_1,A_2,A_3,A_4
> > For k =1,2,3,4, T_k is the triangle with vertices the A_i except A_k; O_k is
the circumcenter of T_k, (O_k) the circumcircle and B_k the isogonal conjugate
of A_k wrt T_k
> > Then the inverse of B_k in (O_k) doesn't depend on k and this point M is the
center of the homothecy mapping O_1,O_2,O_3,O_4 to B_1,B_2,B_3,B_4.
> > Is there a special name for this point M? Do you know some references?
[Chris]
> I do not know a special name for point M.
> However I noticed this. Maybe you know it already.
> Let T(A_1,A_2,A_3,A_4) = Transform A_1,A_2,A_3,A_4 --> O_1,O_2,O_3,O_4.
> Then T^2(A_1,A_2,A_3,A_4) produces a quadrilateral homethetic with
A_1,A_2,A_3,A_4 only rotated 180 degrees. Again Center of Homothecy = M.
> T^4(A_1,A_2,A_3,A_4) produces a quadrilateral homothetic and with same
orientation as A_1,A_2,A_3,A_4.

Thank you for your nice remark.
In fact, if O_1' is the circumcenter of O_2O_3O_4,...,
the same homothecy maps A_i to O_i' and B_i to O_i
The point M is characterized by the angular relations
<A_iMA_j = <A_iA_kA_j +<A_iA_lA_j (oriented angles modulo Pi) where (i,j,k,l) is
any permutation of (1,2,3,4)

Ma question is: how can we prouve the last angular relations? Is there a simple way to prouve it? I try to discover a synthetic proof without success.
Sincerely
Jean-Louis

[Non-text portions of this message have been removed]
• Dear Jean-Louis and other Hyacinthists this follows from two facts : if P,P are inverse in the circumcircle of ABC, then
Feb 25 1 of 2
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Dear Jean-Louis and other Hyacinthists
this follows from two facts :
if P,P' are inverse in the circumcircle of ABC, then
<BPC + <BP'C = 2<BAC
if P,P* are isogonal conjugates wrt ABC, then
<BPC + <BP*C = <BAC
Kind regards. Jean-Pierre

--- In Hyacinthos@yahoogroups.com, Jean-Louis Ayme <jeanlouisayme@...> wrote:
>
> working on a generalization of a Nagel's result, I found in the archive of Hyacinthos the message # 19640
> Dear Chris
> [JP]
> > > Consider a quadrilateral A_1,A_2,A_3,A_4
> > > For k =1,2,3,4, T_k is the triangle with vertices the A_i except A_k; O_k is
> the circumcenter of T_k, (O_k) the circumcircle and B_k the isogonal conjugate
> of A_k wrt T_k
> > > Then the inverse of B_k in (O_k) doesn't depend on k and this point M is the
> center of the homothecy mapping O_1,O_2,O_3,O_4 to B_1,B_2,B_3,B_4.
> > > Is there a special name for this point M? Do you know some references?
> [Chris]
> > I do not know a special name for point M.
> > However I noticed this. Maybe you know it already.
> > Let T(A_1,A_2,A_3,A_4) = Transform A_1,A_2,A_3,A_4 --> O_1,O_2,O_3,O_4.
> > Then T^2(A_1,A_2,A_3,A_4) produces a quadrilateral homethetic with
> A_1,A_2,A_3,A_4 only rotated 180 degrees. Again Center of Homothecy = M.
> > T^4(A_1,A_2,A_3,A_4) produces a quadrilateral homothetic and with same
> orientation as A_1,A_2,A_3,A_4.
>
> Thank you for your nice remark.
> In fact, if O_1' is the circumcenter of O_2O_3O_4,...,
> the same homothecy maps A_i to O_i' and B_i to O_i
> The point M is characterized by the angular relations
> <A_iMA_j = <A_iA_kA_j +<A_iA_lA_j (oriented angles modulo Pi) where (i,j,k,l) is
> any permutation of (1,2,3,4)
>
> Ma question is: how can we prouve the last angular relations? Is there a simple way to prouve it? I try to discover a synthetic proof without success.
> Sincerely
> Jean-Louis
>
> [Non-text portions of this message have been removed]
>
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