- Dear Jean-Pierre

Thanks

[APH]> > Let ABC be a triangle and A'B'C' the orthic triangle.

Variation:

> >

> > 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*.

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

[JPE]

> > A*BC, B*CA, C*AB, resp.

> > The triangles ABC, NaNbNc are perspective.

> > Perspector?

> If U=X(54) is the isogonal conjugate of N and

Corsican..........

> 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)

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

are perspective.

Antreas - [APH]
> > Let ABC be a triangle and A'B'C' the orthic triangle.

[JPE]

> >

> > 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?

> If U=X(54) is the isogonal conjugate of N and

EQUIVALENTLY:

> 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)

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 > [APH]

[JPE]

> > > 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?

>

> > If U=X(54) is the isogonal conjugate of N and

[APH]:

> > 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:

From this we can make this generalization:

>

> 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.

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