## 4891Re: [EMHL] The pedal triangles of Ia, Ib, Ic

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• 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
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