HYACINTHOS 10940

 

 

Dear Milorad, Jean-Pierre and Peter

--- In Hyacinthos@yahoogroups.com, "jpehrmfr" <jean-
pierre.ehrmann@w...> wrote:

>
> Dear Milorad
> [MS]
> > I have found new points on the incircle
> >
> > 1.P1((b-c)^2/(s-a):(c-a)^2/(s-b):(a-b)^2/(s-c)).
> >
> > 2.P2((a(b-c))^2/(s-a):(b(c-a))^2/(s-b):((c(a-b))^2/(s-c)).
> >
> > 3.P3((b+c-2a)^2/(s-a):(c+a-2b)^2/(s-b):(a+b-2c)^2/(s-c)).
> >
> > 4.P4((b^2-c^2)^2/(s-a):(c^2-a^2)^2/(s-b):(a^2-b^2)^2/(s-c)).
> >
> > 5.P5((s-a)[2a^2-a(b+c)+(b-c)^2]:(s-b)[2b^2-b(c+a)+(c-a)^2]:
> > (s-c)[2c^2-c(a+b)+(a-b)^2])
> >
> > This point is the antipode of Feuerbach point Fe=X(11)
> > X(11)=((b-c)^2(s-a):(c-a)^2(s-b):(a-b)^2(s-c))
> > on the Euler or nine-point circle.
> >
> > 6.P6(a^2(b-c)^2(s-a)^3:b^2(c-a)^2(s-b)^3:c^2(a-b)^2(s-c)^3).
>
> Note that if u+v+w=0 then the point u^2/(s-a):v^2/(s-b):w^2/(s-c)
> lies on the incircle.
> There is a typo in P5 : your point lies on the line at infinity

and,

> in any case, the antipode of the Feuerbach point on the 9P-circle
> doesn't lie on the incircle.
> Friendly. Jean-Pierre

Perhaps it is worth noting that the incircle antipode of
Jean-Pierre's point from u:v:w is that from U:V:W with
U = (s-a)(a(v-w)-u(b-c)), etc

In some ways, X(7)-antipodes - points P, Q on incircle with X(7) on PQ
are more attractive (and natural)
X(7) is the perspector of the incircle

the x(7)-antipode of the point from u:v:w is that from U:V:W, where
U = v-w, etc
The corresponding points on the Steiner Inellipse u^2: : and U^2: :
are then X(2)-antipodes.



Best wishes

Wilson Stothers