- 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

> Dear Gilles,

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

>

> [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

> Nikolaos Dergiades.

> 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".

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