[EMHL] Re: Perspective?

Expand Messages
• GENERALIZATION Let ABC be a triangle, Q a point, QaQbQc the pedal triangle of Q, (X) the circumcircle of QaQbQc (=pedal circle of Q), P a point on the OQ line
Message 1 of 23 , Feb 13, 2013
• 0 Attachment
GENERALIZATION

Let ABC be a triangle, Q a point, QaQbQc the pedal triangle of Q,
(X) the circumcircle of QaQbQc (=pedal circle of Q), P a point on
the OQ line and PaPbPc the pedal triangle of P.

Denote:

(X1), (X2), (X3) the reflections of (X) in OPa,OPb,OPc, resp.

A'B'C' = the triangle bounded by the radical axes of ((O),(X1)),((O),(X2)),((O),(X3))

Conjecture:
The triangles A'B'C', X1X2X3 are perspective.

Figure:
http://anthrakitis.blogspot.gr/2013/02/reflecting-pedal-circle.html

Antreas

[APH]
>
> Denote:
>
> PaPbPc = the pedal triangle of a point P on the Euler line
>
> (N1), (N2), (N3) = the reflections of (N) in PaO, PbO, PcO, resp.
>
> A'B'C' = the triangle bounded by the radical axes of
> ((O),(N1)), ((O),(N2)),((O),(N3)).
>
> A'B'C', N1N2N3 are perspective.
>
> True???
• Let HaHbHc be the orthic triangle of ABC, (N1),(N2),(N3) the reflections of the NPC (N) in the sidelines HbHc, HcHa, HaHb of HaHbHc resp. and and A B C the
Message 2 of 23 , Feb 14, 2013
• 0 Attachment
Let HaHbHc be the orthic triangle of ABC, (N1),(N2),(N3) the
reflections of the NPC (N) in the sidelines HbHc, HcHa, HaHb of
HaHbHc resp. and and A'B'C' the triangle bounded by the radical
axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.

Are the triangles ABC, A'B'C' perspective ?

aph

[APH]
> 2. Let HaHbHc be the orthic triangle, (N1),(N2),(N3) the reflections
> of the NPC (N) in the altitudes HHa,HHb,HHc,
> resp. and and A'B'C' the triangle bounded by the radical axes of
> ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
>
> Are the triangles HaHbHc, A'B'C' perspective ?
• Dear Antreas, The triangles ABC, A B C are perspective. Perspector: (Conway notations) (SA^2-3S^2)(S^2-SB^2)(S^2-SC^2))/SA: ... : ... Best regards. Angel M.
Message 3 of 23 , Feb 14, 2013
• 0 Attachment
Dear Antreas,

The triangles ABC, A'B'C' are perspective.

Perspector: (Conway notations)

(SA^2-3S^2)(S^2-SB^2)(S^2-SC^2))/SA: ... : ...

Best regards.
Angel M.

--- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
>
> Let HaHbHc be the orthic triangle of ABC, (N1),(N2),(N3) the
> reflections of the NPC (N) in the sidelines HbHc, HcHa, HaHb of
> HaHbHc resp. and and A'B'C' the triangle bounded by the radical
> axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
>
> Are the triangles ABC, A'B'C' perspective ?
>
> aph
• I am not sure If I have already posted this: Let ABC be a triangle, (N1),(N2),(N3) the reflections of (N)[=NPC] in OA,OB,OC, resp. and A B C the triangle
Message 4 of 23 , Feb 20, 2013
• 0 Attachment
I am not sure If I have already posted this:

Let ABC be a triangle, (N1),(N2),(N3) the reflections of
(N)[=NPC] in OA,OB,OC, resp. and A'B'C' the triangle bounded
by the radical axes of ((O),(N1)),((O),(N2)),((O),(N3)).

Are N1N2N3, A'B'C' perspective?

APH
• One more.... HaHbHc = the orthic triangle. (N1) = the circle (Ha, HaN) ie the circle with center Ha and radius HaN =R/2 Similarly (N2),(N3) A B C = the
Message 5 of 23 , Feb 20, 2013
• 0 Attachment
One more....

HaHbHc = the orthic triangle.
(N1) = the circle (Ha, HaN) ie the circle with center Ha and radius HaN =R/2
Similarly (N2),(N3)

A'B'C' = the triangle bounded by the radical axes of
((O),(N1)),((O),(N2)),((O),(N3)).

Are the triangles HaHbHc, N1N2N3 perspective?

aph

On Wed, Feb 20, 2013 at 1:29 PM, Antreas <anopolis72@...> wrote:

> **
>
>
> I am not sure If I have already posted this:
>
> Let ABC be a triangle, (N1),(N2),(N3) the reflections of
> (N)[=NPC] in OA,OB,OC, resp. and A'B'C' the triangle bounded
> by the radical axes of ((O),(N1)),((O),(N2)),((O),(N3)).
>
>
> Are N1N2N3, A'B'C' perspective?
>
> APH
>
> __
>

[Non-text portions of this message have been removed]
• Let ABC be a triangle with excenters Ia,Ib,Ic. The NPC of AHIa interscts the excircle (Ia) at A other than the Feuerbach point Fa. The NPC of BHIb
Message 6 of 23 , May 6, 2014
• 0 Attachment

Let ABC be a triangle with excenters Ia,Ib,Ic.

The NPC of  AHIa interscts the excircle (Ia) at A' other than the
Feuerbach point Fa.

The  NPC of  BHIb intersects the excircle (Ib) at B' other than the
Feuerbach point Fb,

The  NPC of  CHIc intersects the excircle (Ic) at C' other than the
Feuerbach point Fc.

Are the triangles ABC, A'B'C' perspective?

In any case, has the triangle A'B'C' any interesting properties?

APH

Your message has been successfully submitted and would be delivered to recipients shortly.