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• ## Re: A CONJECTURE on NEUBERG (was: Thomson ?)

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• Dear Antreas and other Hyacinthists, [APH] ... at oo. ... the ... Unfortunately, I don t think so. I think that the locus for the Euler line is a quartic and
Message 1 of 2 , Sep 1, 2001
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Dear Antreas and other Hyacinthists,

[APH]
>
> Another recent conjecture of mine is this
> (which, I hope isn't false!):
>
> Let ABC be a triangle, and A'B'C' the pedal triangle of P.
>
>
> A
> /\
> / \
> / \
> C' B'
> / \
> / P \
> Ab Ac
> / \
> B-------A'-------C
>
> Let Ab, Ac be the orth. proj. of A' on AB, AC resp.
> Bc, Ba " B' BC, BA
> Ca, Cb " C' CA, CB
>
> Conjecture:
> The locus of P such that the Euler Lines of the triangles
> A'AbAc, B'BcBa, C'CaCb are concurrent is Neuberg cubic + (O) + Line
at oo.
> The same is true for the Brocard axes instead of the Euler lines.
> The point of concurrence lies on the Euler line / Brocard Axis of
the
> pedal triangle A'B'C'.
>
> Is this conjecture true?

Unfortunately, I don't think so.
I think that the locus for the Euler line is a quartic and for the
Brocard line a curve of degree 10.
I didn't find any classical center on the second one and only H and
the far-out oint on the first one.

>
> Or, at least, is it true that the OH/OK lines of the triangles
> A'AbAc, B'BcBa, C'CaCb concur [at the OH/OK line of A'B'C']
> for P in {H, O, I} ?

You're right for P = H and the Eulerline : the four Euler lines
concur.
Friendly. Jean-Pierre
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