Dear Jean-Pierre,

[APH]:

>> Let ABC be a triangle, and A'B'C' the pedal triangle of P.

>>

>>

>> A

>> /\

>> / \

>> / \

>> C' B'

>> / \

>> / P \

>> Ab Ac

>> / \

>> B-------A'-------C

>>

>> Let Ab, Ac be the orth. proj. of A' on AB, AC resp.

>> Bc, Ba " B' BC, BA

>> Ca, Cb " C' CA, CB

>> [...] is it true that the OH/OK lines of the triangles

>> A'AbAc, B'BcBa, C'CaCb concur [at the OH/OK line of A'B'C']

>> for P in {H, O, I} ?

[JPE]:

>You're right for P = H and the Eulerline : the four Euler lines

>concur.

So, we have the Theorem:

Let AA', BB', CC' be the three altitudes of ABC, and

Let Ab, Ac be the orth. proj. of A' on AB, AC resp.

Bc, Ba " B' BC, BA

Ca, Cb " C' CA, CB

Then the Euler lines of A'B'C', A'AbAc, B'BcBa, C'CaCb are concurrent.

Which is the point of concurrence [a point lying on the Euler line of

the orthic triangle A'B'C' of ABC]? Is it in ETC?

And another conjecture:

The Euler lines of the triangles AAbAc, BBcBa, CCaCb are concurrent.

[are also concurrent for P = I or O ?]

Thank You!

Antreas