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## 21052Re: Simson lines

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• Jun 6, 2012
Yes, it is a particular case of S-triangles.

Two triangles ABC and A'B'C' inscribed in the same circle are S-triangles iff arc(AA')+arc(BB')+arc(CC')=0 (mod 2PI).
The Simson lines of A',B',C', with respect to ABC and the Simson lines of A,B,C, with respect to A'B'C' are concurrent in the midpoint of [HH'] where H and H' are the orthocenters of ABC and A'B'C'.

The triangles CAB and CPQ are S-triangles iff PQ is parallel to AB.
So, the Simson lines of P,Q, and C are concurrent. But the Simson line of C is the altitude from C.

See
[1] Traian Lalescu, A Class of Remarkable Triangles,
Gazeta Matematica , vol XX, feb. 1915 ,p 213
(in Romanian)

[2] Trajan Lalesco, La geometrie du triangle ,Bucharest,
1937, Paris, Libraire Vuibert

Sincerely
Daniel Vacaretu

--- In Hyacinthos@yahoogroups.com, "Alexey Zaslavsky" <zasl@...> wrote:
>
> Dear colleagues!
> Is next fact known?
> Let P, Q be two point on the circumcircle of triangle ABC. Then their Simson lines meet on the altitude from C iff PQ is parallel to AB.
>
> Sincerely Alexey
>
> [Non-text portions of this message have been removed]
>
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