> 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
> 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
> 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
> on B'C' lies on AP[A] and so on.
> With regard to I), I would like to know if this transformation has
> studied at all and perhaps even has a name. Regarding II), I am
> 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
The transformation is rational but not birational (in fact there are
7 points P -not necessarily real- with a given image P')