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and 
. Prove that
Where 
satisfies 
. If 
, find the maximum possible value of 
.
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
, two tangents are drawn touching at points 
and 
. A point 
is chosen on the segment 
. Let points 
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects again at point 
(
). Prove that the line 
passes through the centroid of triangle 
.
, 
is a point on 
and 
is a point on 
such that 
and 
are bisectors of 
and 
respectively. Let 
and be the feet of perpendiculars from the midpoint of 
onto 
and , respectively. Prove that 
.
![$ In \ my \ solution \ |PQ|=|PM|+|PN| \ is \ true \ (\forall) P\in [DE].$](http://latex.artofproblemsolving.com/8/1/1/81137de4aa9e622673e4fbe9ec20d7e2c1a1f84e.png)
to 
be 
and the foot of perpendicular from to be 
and the foot of the perpendicular from to be 
. Notice that because and are angle bisectors of and 
respectively, we have that 
and similarly for the other, so this gives that 
and 
. From here, it is trivial to finish, just note that 
which gives the desired.
![$ In \ barycentric \ coordonate \ have: \ D(a;b;0), E(a;0;c) \Rightarrow DE \ equation \ is: \\ \frac{x}{a}=\frac{y}{b}+\frac{z}{c} \Leftrightarrow
\frac{S_{PBC}}{a}=\frac{S_{PAC}}{b}+\frac{S_{PAC}}{c} \Leftrightarrow |PQ|=|PM|+|PN| \ (if \ P\in[DE]).$](http://latex.artofproblemsolving.com/d/7/f/d7fdf2b6a608a4fc78bfdd5272d8ad7b6ccb32b5.png)
.
in 
and in 
as distance of point to a line is such. So it is sufficient to prove that for two distinct points 
.
QED.