HYACINTHOS 1498

Dear Fred, Jean-Pierre and all,
[ Fred]

>> Find the locus (1) of points P such that La', Lb' and Lc' concur, find
>> the locus (2) of the point Q of concurence, describe the transformation
>> f: P->Q.
>>
>> I have found the following results:
>> 1) the locus (1) is a cubic curve, and the result of Parry is that this
>> cubic has only one asymptote, parallel to the Euler line.
>> 2) the locus (2) is a cubic curve, and the Parry'point is on this curve.

I think there's some confusion here between the Parry point (X111) and the
Parry reflection point (X399), the latter being on the Neuberg cubic.

>> 3) the transformation f is a quadratic transformation with A,B,C and H
>> fixed.
>> 4) the locus (2) is the Neuberg cubic.
>> 5) The following points are on the locus (1): ABCH, cyclic points, point
>> at infinity on the Euler line, isogonic points.
>> Total: 9 points, but this locus is not the Neuberg cubic, for exemple
>> the incenter I is not on the locus.
>> But these points are on the Neuberg: contradiction??? Help!!!

[JP]

> It is true that the locus (2) is the Neuberg cubic, but this curve cannot go
> through the Parry point - the 6 common points with the circumcircle are
> A,B,C,the cyclic points and the isogonal conjugate of the infinite point of
> the Euler line -
>
> The locus (1) - not self-isogonal - and the Neuberg cubic generate a pencil
> of cubics - that's why they have 9 common points -
> All those cubics are circular with singular focus on the line GM - G
> =centroid, M = midpoint of OK.
> This line intersects the circumcircle at F = focus of Kiepert's parabola =
> focus of Neuberg cubic and T = Tarry point,.
> The focus of (1) should be 3 G - 2 F.

May I add several small comments on this cubic I'm used to call (Kn) :
1) N=X5, X265= X186* where X186 is the inverse of H in the circumcircle, the
traces of X265 are 5 points on it.
2) the tangents in A,B,C are the altitudes.
3) the tangents to Neuberg in I,Ia,Ib,Ic are tangent to (Kn) and are // to
the Euler line.
4) the real asymptote is the // to the Euler line at X399.
5) the fourth intersection with the circumcircle is the isogonal of the
infinite point of the Euler line of the orthic triangle.
6) I remember there's a very easy way to tranform Neuberg into (Kn) but I
can't find my old notes : there's something to do with circles through O &
H.

What is its intersection with the real asymptote ?

BTW, and totally by chance, I realised the symmetric of K about N is on the
line through G, X182=OK midpoint, X98=Tarry, X110.
The coordinates are not too ugly and I think it deserves a name in Clark's
ETC.

Best regards

Bernard Gibert