- 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]> > Consider a quadrilateral A_1,A_2,A_3,A_4

the circumcenter of T_k, (O_k) the circumcircle and B_k the isogonal conjugate

> > For k =1,2,3,4, T_k is the triangle with vertices the A_i except A_k; O_k is

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.

A_1,A_2,A_3,A_4 only rotated 180 degrees. Again Center of Homothecy = 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

> 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

<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:

>

> 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]

> > > 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]

>