## [EMHL] Re: axes

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• Dear Steve [Steve] ... parallel ... [JP] ... asymptots of ... precevian ... the ... [Steve] ... Because they have a diagonal matrix?? Let L1,L2 be the infinite
Message 1 of 5 , Mar 4, 2007
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Dear Steve

[Steve]
> > > Circumconics whose perspectors are collinear with K have
parallel
> > > axes. I cannot find a similar relation for inconics. What is the
> > > criterion for inconics to have parallel axes?

[JP]
> > The axes of the inconic with center M are parallel to the
asymptots of
> > the conjugate rectangular hyperbola going through M (ie the
> > rectangular
> > hyperbola through M,incenter, excenters,the vertices of the
precevian
> > triangle of M,...).
> > Hence, two inconics have parallel axis iff their centers lie on
the
> > same conjugate rectangular hyperbola

[Steve]
> Thanks. I used to call these "polar conics" but Conway has me
> calling them "diagonal conics." Terminology...ug!

Because they have a diagonal matrix??

Let L1,L2 be the infinite points of the diagonal rectangular
hyperbola h going though M and co the inconic with center M.
As ABC is selfpolar wrt h and circumsribed in co, P is vertex of a
triangle inscribed in h and selfpolar wrt co.
As P is the pole of Linf wrt co, this triangle is necessarily PL1L2;
which means that PL1, PL2 are conjugate diameters of co. As they are
perpendicular, they are the axes of co.
Friendly. Jean-Pierre
• ... Yes, I think this is what they were called when John was at Cambridge, but it is strange terminology for him has he does not like geometry terms whose
Message 2 of 5 , Mar 4, 2007
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On Mar 4, 2007, at 11:08 AM, jpehrmfr wrote:

> [Steve]
> > Thanks. I used to call these "polar conics" but Conway has me
> > calling them "diagonal conics." Terminology...ug!
>
> Because they have a diagonal matrix??

Yes, I think this is what they were called when John was at
Cambridge, but it is strange terminology for him has he does not like
geometry terms whose origin is not geometrical.

>
> Let L1,L2 be the infinite points of the diagonal rectangular
> hyperbola h going though M and co the inconic with center M.
> As ABC is selfpolar wrt h and circumsribed in co, P is vertex of a
> triangle inscribed in h and selfpolar wrt co.
> As P is the pole of Linf wrt co, this triangle is necessarily PL1L2;
> which means that PL1, PL2 are conjugate diameters of co. As they are
> perpendicular, they are the axes of co.

Very pretty proof from which I learned a lot

Thanks and best regards,

Steve

Notation:
20web/notation.html

Triangle web page:
http://paideiaschool.org/TeacherPages/Steve_Sigur/geometryIndex.htm

Other math:
http://paideiaschool.org/TeacherPages/Steve_Sigur/interesting2.htm

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