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## Re: [EMHL] Adams' circle

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• Dear Nikos [ND] ... This should be known by Conway s extraversion BTW, and Conway s Circle is also centered at I and has three brother-circles by extraversion.
Message 1 of 2 , Jul 4, 2005
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Dear Nikos

[ND]
>it is known http://mathworld.wolfram.com/AdamsCircle.html
>that the parallel lines from the Gergonne point
>Ge = ( 1/(s-a) , 1/(s-b), 1/(s-c) ) in barycentrics
>of triangle ABC to the sides of its cevian triangle meet the
>sides of ABC at six points that are lying on Adams' circle
>a circle with center the incenter of ABC.
>
>Is it known that the same property holds also for the other
>Gergonne points Ga, Gb, Gc of triangle ABC?

This should be known by Conway's extraversion

BTW, and Conway's Circle is also centered at I and has
three brother-circles by extraversion.

APH
--
• Thanks Antreas. [ND] ... [APH] ... If the point P has barycentrics (x : y : z) and A B C is its cevian triangle then the parallel line from P to B C meets
Message 2 of 2 , Jul 5, 2005
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Thanks Antreas.

[ND]
> >it is known http://mathworld.wolfram.com/AdamsCircle.html
> >that the parallel lines from the Gergonne point
> >Ge = ( 1/(s-a) , 1/(s-b), 1/(s-c) ) in barycentrics
> >of triangle ABC to the sides of its cevian triangle meet the
> >sides of ABC at six points that are lying on Adams' circle
> >a circle with center the incenter of ABC.
> >
> >Is it known that the same property holds also for the other
> >Gergonne points Ga, Gb, Gc of triangle ABC?

[APH]
> This should be known by Conway's extraversion
>
> BTW, and Conway's Circle is also centered at I and has
> three brother-circles by extraversion.

If the point P has barycentrics (x : y : z)
and A'B'C' is its cevian triangle then the parallel line
from P to B'C' meets the sides AC, AB of triangle ABC
at the points Ab, Ac. Similarly define the points Bc,Ba, Ca, Cb.
The six points are lying on a conic with center the point
( x(y+z) : y(z+x) : z(x+y) ) in barycentrics that is the
complement of the isotomic conjugate of P.
If the point P is inside the Steiner circumellipse the conic
is an ellipse or circle for the points Go, Ga, Gb, Gc.
If the point P is on the Steiner circumellipse the conic
is a parabola.
If the point P is outside of the Steiner circumellipse the conic
is an hyperbola and if P is also on the conic
xx + yy + zz + 3xy + 3yz + 3zx = 0
the Steiner circumellipse of the anticomplement triangle of ABC
the points Ab, Bc, Ca are collinear and the points Ac, Ba, Cb
are also collinear.

Best regards
Nikolaos Dergiades
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