[/quote]
,
and 
.
Where 
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
,
and 
.
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
and let 
be the foot of the
and 
in line 
and 
respectively. A line through 
while 
and 
and 
are tangent to each other.
with circumcircle 
.
is the angle bisector of 
.
intersects 
.
angle bisector of 
if and only if 
.
be real numbers satisfying
.
,
past 
.
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
and 
intersect 
and 
at 
intersect 
.
intersects 
at 
intersect 
.
intersects 
and 
intersect on the circumcircle of triangle 
.
.![$D \in [AB]$](http://latex.artofproblemsolving.com/0/7/9/07914568284f3822cf15730a6a0e868f0bd79996.png)
is a point. Perpendiculars from ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
and ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
intersects with ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
intersects with 
and 
.![$$[E_1F_1]=\frac{3}{4}$$](http://latex.artofproblemsolving.com/3/e/f/3efdf53cb3ff4b2a7fa60691f5aac9b3eeecf00c.png)
be the point on 
.
Y by
Let 
be an acute, non isosceles triangle and 
be its circumcircle (with center 
). Denote by 
the centroid of the triangle , by 
the foot of the altitude from 
onto the side 
and by 
the midpoint of 
. The line 
intersects at 
.
1. Prove that
.
2. The ray
intersects at L. Denote by 
the circumcenter of the triangle 
. Prove that 
and 
intersect on the circle .
be an acute, non isosceles triangle and 
be its circumcircle (with center 
). Denote by 
the centroid of the triangle , by 
the foot of the altitude from 
onto the side 
and by 
the midpoint of 
. The line 
intersects at 
.1. Prove that
.2. The ray
intersects at L. Denote by 
the circumcenter of the triangle 
. Prove that 
and 
intersect on the circle .
(isotomic conjugate of orthocenter) has barycentric coordinates 
.
.
and 
.
are colinear.
,
