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, 
. The internal angle bisector of 
and the external angle bisector of intersect the ray 
at points 
and 
, respectively. Given that 
, prove that 
.
![\[f(xf(y) + x) = xy + f(x)\]](http://latex.artofproblemsolving.com/9/9/f/99f580ebc50846e6bc2c004667559922749a4dfa.png)
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
be an acute triangle. Let 
be the altitude on , and let 
be any interior point on . Lines 
, when extended, intersect 
at 
respectively. Prove that 
.
and 
are given. The lines 
and 
meet at 
and the lines parallel to them and passing through 
and 
meet at 
. The points 
, 
are defined similarly. Prove that 
are either parallel or concurrent.
(possibly at infinity) with the same barycentric coordinates wrt and 
. be a point (not at infinity) and 
, 
two parallel lines. Consider an affine transform fixing and let 
, 
be the images of and . Then , 
, 
are collinear. lies on line 
, the other two lines being similar. Suppose is not at infinity. By the definition of , there exists an affine transform fixing taking 
; it also sends lines 
and 
. By the lemma , 
, 
are collinear, as desired. at infinity can be settled with a continuity argument.
is the locus of all points 
such that![$$ [XAB] \cdot [A'B'C'] = [XA'B'] \cdot [ABC] $$](http://latex.artofproblemsolving.com/6/7/d/67d66920e5be0a21379c7b269ee3b0a3465a930e.png)
where areas are directed and ![$[\lambda]$](http://latex.artofproblemsolving.com/0/f/1/0f16657b1125c41a1cc64851eb0089560e0b2fd7.png)
denotes the area of figure 
. Then it is not hard to see that lines 
concur (like the intersection of any two lines must lie on the third also). 
satisfying![\[x(A - A') + y(B - B') + z(C - C') = 0.\]](http://latex.artofproblemsolving.com/9/2/a/92a9a9d90ba5ebd8fd33a02013e227f2c14641cd.png)
Defining 
and similar, consider the underdetermined system![\[
\begin{bmatrix}
p_a & p_b & p_c\\
q_a & q_b & q_c
\end{bmatrix}
\cdot
\begin{bmatrix}
x \\ y \\ z
\end{bmatrix}
=
\begin{bmatrix}
0 \\ 0
\end{bmatrix}.
\]](http://latex.artofproblemsolving.com/5/c/b/5cb2005f70ec9d07d979239e51799caabc992e55.png)
If the rows of the 
matrix are proportional then 
,
(up to scaling), which are the desired coordinates.
and 
.
and 
or vice versa, implying the collinearity.