- I'm sorry. Of course, the condition P=f(a),f(b),f(c) must be nessesary for

that construction, i.e. it is valid not for any central point.

Yours,

Alex----- Original Message -----

From: "alex" <alex_geom@...>

To: <Hyacinthos@yahoogroups.com>

Sent: Tuesday, May 13, 2003 12:19 AM

Subject: Re: [EMHL] New point - New mapping?

> Dear Milorad!

> It seems to me that we can get some generalization of that

construction -if

> I dont't mistake, that fact is correct for any central point - so we have

> something like central mapping.(And, for me, it was rather unexpected).

> More precisely, let P be the point with barycentric f(a), f(b), f(c)

> (nothing particular would change in case f(a,b,c), f(b,c,a),f(c,b,a)).

> AP=la,BP=lb,CP=lc, S=f(a)+f(b)+f(c).

> Let Pa is the point with barycentric f(a),f(lc),f(lb) - i.e. the point

> similar to P, but wrt triangle PBC.

> Pb and Pc let us define in the same way.

> So, (if I don't mistake in calculations) APa,BPb,CPc have common point

witn

> barycentric 1/f(b)f(c)+f(la)S,1/f(a)f(c)+f(lb)S,1/f(a)f(b)+f(lc)S - wrt.

> triangle ABC.

> Does that fact well-known?

> Or may be it is all an illusion?

> Best regards,

> Yours sincerely,

> Alex

> ----- Original Message -----

> From: "Milorad Stevanovic" <milmath@...>

> To: <Hyacinthos@yahoogroups.com>

> Sent: Monday, May 12, 2003 6:02 PM

> Subject: [EMHL] New point

>

>

> > Dear Alexey,

> > 1.If I1,I2,I3 are the incenters of triangles

> > BIC,CIA,AIB with I as incenter of ABC,

> > then AI1,BI2,CI3 are concurent in point

> > M(a/(1+2cos(A/2)):b/(1+2cos(B/2)):

> > c/(1+2cos(C/2))).( in barrycentrics).

> > Best regards

> > Sincerely

> > Milorad R.Stevanovic

> >

> >

> > [Non-text portions of this message have been removed]

> >

> >

> >

> >

> >

> > Your use of Yahoo! Groups is subject to

http://docs.yahoo.com/info/terms/

> >

> >

> >

>

>

>

>

>

> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

>

>

> >I'm sorry. Of course, the condition P=f(a),f(b),f(c) must be nessesary for

Dear Alex, Milorad and other colleagues!

>that construction, i.e. it is valid not for any central point.

>Yours,

>Alex

>> Dear Milorad!

>> It seems to me that we can get some generalization of that

>construction -if

>> I dont't mistake, that fact is correct for any central point - so we have

>> something like central mapping.(And, for me, it was rather unexpected).

>> More precisely, let P be the point with barycentric f(a), f(b), f(c)

>> (nothing particular would change in case f(a,b,c), f(b,c,a),f(c,b,a)).

>> AP=la,BP=lb,CP=lc, S=f(a)+f(b)+f(c).

>> Let Pa is the point with barycentric f(a),f(lc),f(lb) - i.e. the point

>> similar to P, but wrt triangle PBC.

>> Pb and Pc let us define in the same way.

>> So, (if I don't mistake in calculations) APa,BPb,CPc have common point

>witn

>> barycentric 1/f(b)f(c)+f(la)S,1/f(a)f(c)+f(lb)S,1/f(a)f(b)+f(lc)S - wrt.

>> triangle ABC.

I think, that a condition P=(f(a),f(b),f(c)) is not nessesary. For example

it seems, that this hypothesis is correct for the circumcenter O. But it

isn't correct for Gergonne's point G. It is interesting to formulate the

nessesary and sufficient conditions, but I don't know how to do it.

Sincerely your Alexey> >I'm sorry. Of course, the condition P=f(a),f(b),f(c) must be nessesary

for

> >that construction, i.e. it is valid not for any central point.

Dear Alexey!

> >Yours,

> >Alex

> Dear Alex, Milorad and other colleagues!

> I think, that a condition P=(f(a),f(b),f(c)) is not nessesary. For example

> it seems, that this hypothesis is correct for the circumcenter O. But it

> isn't correct for Gergonne's point G. It is interesting to formulate the

> nessesary and sufficient conditions, but I don't know how to do it.

>

> Sincerely your Alexey

When I've wrote about "necessaty", I wanted to use that therm not in

mathematical sence, but only to stress that my conclusion is valid wrt the

points with f(a),...,... coordinates. Of course, there exsist some central

points of more general type (cirumcenter, for example, as You have mentioned

it), where condition of intersection still remains

true. But I hardly believe, that it's possible to explore general case.

Best regards,

Sincerely Yours,

Alex>

>

>

>

>

>

> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

>

>

>>Dear Alexey!

mentioned

>When I've wrote about "necessaty", I wanted to use that therm not in

>mathematical sence, but only to stress that my conclusion is valid wrt the

>points with f(a),...,... coordinates. Of course, there exsist some central

>points of more general type (cirumcenter, for example, as You have

>it), where condition of intersection still remains

There is once more interesting problem. Let P is any center, P' - a point of

>true. But I hardly believe, that it's possible to explore general case.

>Best regards,

>Sincerely Yours,

>Alex

>

intersection of APa, BPb, CPc. When P'=P. I found 4 points:

centroid M, orthocenter H and 2 Torrichelli's points T1, T2.

Alexey