Dear Eisso

> as I was playing with some Cevian triangles I noticed the following:

>

> I) Let A'B'C' be the Cevian triangle of a point P w.r.t. a

triangle

> ABC. Now let P[A] be the reflection of P in B'C' and P[B] and P[C]

> defined analogously. Then AP[A], BP[B], and CP[C] are concurrent

in a

> point P', thus defining a (birational) transformation P to P'.

>

> II) The pedal points of the altitudes of A'B'C' each lie on the

> corresponding cevian XP[X], i.e. the pedal point of the altitude

from A'

> on B'C' lies on AP[A] and so on.

>

> With regard to I), I would like to know if this transformation has

been

> studied at all and perhaps even has a name. Regarding II), I am

just

> wondering if this property is known.

Look at #8039; you'll see a proof of your results (your point P' is

the isogonal conjugate of P wrt the orthic triangle of A'B'C')

Clark Kimberling named P->P' the Orion transformation and the first

result above was named the Begonia theorem by Darij Grinberg who gave

a proof available at

http://de.geocities.com/darij_grinberg/Begonia.zip
The transformation is rational but not birational (in fact there are

7 points P -not necessarily real- with a given image P')

Friendly. Jean-Pierre