with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
be a positive integer. A positive integer 
is called a benefactor of 
and 
such that 
.
is a benefactor of 
because 
.
benefactors.
and 
: Prove that :
Y by Adventure10, Mango247, and 1 other user
Let 
be an acute triangle. Find the locus of the points 
, in the interior of 
, such that 
, where 
and 
are the feet of the perpendiculars from to the lines 
and 
, respectively.
be an acute triangle. Find the locus of the points 
, in the interior of 
, such that 
, where 
and 
are the feet of the perpendiculars from to the lines 
and 
, respectively.
.
:
.
be the isogonal of 
,
lies on 
.
![$\frac{1}{2}(AB)(CD')=[ACD']+[BCD']$](http://latex.artofproblemsolving.com/7/0/c/70ca3c1d3d82124fb872ab9e68e9b66a08fd6202.png)
![$\frac{1}{2}(AB)(CD')=[ABC]$](http://latex.artofproblemsolving.com/a/0/0/a003bf6ef0076f4ae2257fc4406c2b0794a788dc.png)
.
such that 
,
is the circumcentre of the triangle. (because circumcentre, orthocentre are isogonal conjugates)
