## Re: [EMHL] New point - New mapping?(correction)

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• 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 ... From:
Message 1 of 4 , May 12 2:57 PM
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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?

> 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 -----
> 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
> >
> >
> > [Non-text portions of this message have been removed]
> >
> >
> >
> >
> >
> > Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
> >
> >
> >
>
>
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>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>
• ... 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
Message 2 of 4 , May 13 6:01 AM
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>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
>> 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.
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.

• ... for ... Dear 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
Message 3 of 4 , May 13 11:35 AM
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> >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
> 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.
>
Dear 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
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> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
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• ... mentioned ... There is once more interesting problem. Let P is any center, P - a point of intersection of APa, BPb, CPc. When P =P. I found 4 points:
Message 4 of 4 , May 13 10:03 PM
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>Dear 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
>
There is once more interesting problem. Let P is any center, P' - a point of
intersection of APa, BPb, CPc. When P'=P. I found 4 points:
centroid M, orthocenter H and 2 Torrichelli's points T1, T2.

Alexey
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