## Re: [EMHL] The pedal triangles of Ia, Ib, Ic

Expand Messages
• Dear Antreas, Paul and Nik, [ND]: The Euler lines of the pedal triangles of the excenters Ia,Ib,Ic are concurrent. If my computation is (at least this time)
Message 1 of 35 , Mar 1, 2002
Dear Antreas, Paul and Nik,

[ND]: "The Euler lines of the pedal triangles of
the excenters Ia,Ib,Ic are concurrent."

If my computation is (at least this time) correct, the only OX concuring
lines are the lines OG (Euler) and OK (Brocard).

Best regards

Bernard
• Dear Antreas ... I apologize, you are right.. I am mistaked in my notations. Gilles
Message 35 of 35 , Mar 3, 2002
Dear Antreas

> Dear Gilles,
>
> [APH]
> Let IaaIabIac be the pedal triangle of Ia.
>
> A' := IaIaa /\ IabIac
>
> Similarly B', C'
>
> The triangles ABC, A'B'C' are perspective.
>
> [...]
>
>
> ==> the point of concurrence is the centroid G.
>
> [...]
>
> 1. A" := the orth. projection of Iaa on IabIac.
>
> Similarly B", C"
>
> Then the triangles ABC, A"B"C" are perspective.
>
> [GB]:
> The triangle MaMbMc with sides IabIac, IbaIbc, IcaIcb is homothetic to
> IaIbIc. A' is the feet on IabIac of the altitude, as we prove with
> A'B'C' is the orthic triangle of MaMbMc, and ABC is the orthic triangle
> of IaIbIc. So ABC and A'B'C' are homothetic, therefore they are
> perspective. Perspector is the Mittelpunkt X_9 of ABC.
>
> *********
>
> I guess your A'B'C' is mine A"B"C".

I apologize, you are right.. I am mistaked in my notations.

Gilles

>
>
> Let's see what the perspector of ABC, A"B"C" is:
>
>
> A
> ..................
> / \
> / \
> B----A1-Iaa--------C
> / \
> A3 A2
> Iac \
> / A" Iab
> / \
> Ia
>
> [...]
>
> ==> AA" is the a-cevian of Mittenpunkt (cot(A/2) ::)
>
>
> Antreas
Your message has been successfully submitted and would be delivered to recipients shortly.