## Re: A configuration

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• Dear Jean-Pierre Thanks [APH] ... Variation: Lab := The Reflection of BB in AA Mab := The Parallel to Lab through B (instead of C). Lac := The Reflection of
Message 1 of 5 , Jun 1, 2009
Dear Jean-Pierre

Thanks

[APH]
> > Let ABC be a triangle and A'B'C' the orthic triangle.
> >
> > Denote:
> >
> > Lab := The Reflection of BB' in AA'
> > Mab := The Parallel to Lab through C.
> >
> > Lac := The Reflection of CC' in AA'
> > Mac := The Parallel to Lac through B.
> >
> > A* := Mab /\ Mac
> >
> > Similarly B*,C*.

Variation:

Lab := The Reflection of BB' in AA'
Mab := The Parallel to Lab through B (instead of C).

Lac := The Reflection of CC' in AA'
Mac := The Parallel to Lac through C (instead of B).

A* := Mab /\ Mac

Similarly B*,C*.

A*B*C* is the circumcevian triangle of H.

[APH]
> > 5. Let Na, Nb, Nc be the NPC centers of the triangles
> > A*BC, B*CA, C*AB, resp.
> > The triangles ABC, NaNbNc are perspective.
> > Perspector?

[JPE]
> If U=X(54) is the isogonal conjugate of N and
> V=X(140)=midpoint(ON), the perspector is P=-3U+4V
> (this point lies on the line through the Lemoine point and
> the two Corsican Imperial points)

Corsican..........

ÄéåôÞñåé áßìáôüò tïõ
åéò ôáò öëÝâáò ôïõ ñáíßäá
ï Êïñóéêáíüò ï Ý÷ùí
ôïí Ôáàãåôïí Ðáôñßäá

In the variation, we have: A'B'C'[=orthic], NaNbNc
are perspective.

Antreas
• [APH] ... [JPE] ... EQUIVALENTLY: Let ABC be a triangle, A1B1C1 the circumcevian triangle of H, and M1M2M3 the Medial triangle of ABC. Let A*,B*,C* be the
Message 2 of 5 , Jun 3, 2009
[APH]
> > Let ABC be a triangle and A'B'C' the orthic triangle.
> >
> > Denote:
> >
> > Lab := The Reflection of BB' in AA'
> > Mab := The Parallel to Lab through C.
> >
> > Lac := The Reflection of CC' in AA'
> > Mac := The Parallel to Lac through B.
> >
> > A* := Mab /\ Mac
> >
> > Similarly B*,C*.

> > 5. Let Na, Nb, Nc be the NPC centers of the triangles
> > A*BC, B*CA, C*AB, resp.
> > The triangles ABC, NaNbNc are perspective.
> > Perspector?

[JPE]
> If U=X(54) is the isogonal conjugate of N and
> V=X(140)=midpoint(ON), the perspector is P=-3U+4V
> (this point lies on the line through the Lemoine point and
> the two Corsican Imperial points)

EQUIVALENTLY:

Let ABC be a triangle, A1B1C1 the circumcevian triangle
of H, and M1M2M3 the Medial triangle of ABC.

Let A*,B*,C* be the reflections
of A1,B1,C1 in M1,M2,M3, resp. (symmetric points)

Let Na, Nb, Nc be the NPC centers of the triangles
A*BC, B*CA, C*AB, resp.
The triangles ABC, NaNbNc are perspective.

APH
• ... [JPE] ... From this we can make this generalization: Let ABC be a triangle, A1B1C1 the circumcevian triangle of point P, and M1M2M3 the Medial triangle of
Message 3 of 5 , Jun 3, 2009
> [APH]
> > > Let ABC be a triangle and A'B'C' the orthic triangle.
> > >
> > > Denote:
> > >
> > > Lab := The Reflection of BB' in AA'
> > > Mab := The Parallel to Lab through C.
> > >
> > > Lac := The Reflection of CC' in AA'
> > > Mac := The Parallel to Lac through B.
> > >
> > > A* := Mab /\ Mac
> > >
> > > Similarly B*,C*.
>
> > > 5. Let Na, Nb, Nc be the NPC centers of the triangles
> > > A*BC, B*CA, C*AB, resp.
> > > The triangles ABC, NaNbNc are perspective.
> > > Perspector?
>
[JPE]
> > If U=X(54) is the isogonal conjugate of N and
> > V=X(140)=midpoint(ON), the perspector is P=-3U+4V
> > (this point lies on the line through the Lemoine point and
> > the two Corsican Imperial points)

[APH]:
>EQUIVALENTLY:
>
> Let ABC be a triangle, A1B1C1 the circumcevian triangle
> of H, and M1M2M3 the Medial triangle of ABC.
>
> Let A*,B*,C* be the reflections
> of A1,B1,C1 in M1,M2,M3, resp. (symmetric points)
>
> Let Na, Nb, Nc be the NPC centers of the triangles
> A*BC, B*CA, C*AB, resp.
> The triangles ABC, NaNbNc are perspective.

From this we can make this generalization:

Let ABC be a triangle, A1B1C1 the circumcevian triangle
of point P, and M1M2M3 the Medial triangle of ABC.

Let A*,B*,C* be the reflections
of A1,B1,C1 in M1,M2,M3, resp. (symmetric points)

Let Na, Nb, Nc be the NPC centers of the triangles
A*BC, B*CA, C*AB, resp.

Which is the locus of P such that:
The triangles ABC, NaNbNc are perspective ?
(O,I,H lie on the locus)

Also:
Which is the locus of P such that ABC, A*B*C*
are perspective (or orthologic)?

Which is the locus of P such that ABC, OaObOc
are perspective?
(Oa,Ob,Oc = Circumcenters of A*BC, B*CA, C*AB)

Antreas
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