[/quote]
,
and 
.
Where 
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
,
and 
.
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
be an acute-angled triangle with 
and let 
be the foot of the
-
.
and 
in line 
and 
respectively. A line through 
and 
respectively such that 
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 
.
with orthocenter 
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)
and 
.
be the point on 
.
Y by Adventure10, Mango247
A non-isosceles triangle 
has sides 
, 
, 
with the side 
lying opposite to the vertex 
. Let 
be the midpoint of the side , and let 
be the point where the inscribed circle of triangle touches the side . Denote by 
the reflection of the point in the interior angle bisector of the angle . Prove that the lines 
, 
and 
are concurrent.
has sides 
, 
, 
with the side 
lying opposite to the vertex 
. Let 
be the midpoint of the side , and let 
be the point where the inscribed circle of triangle touches the side . Denote by 
the reflection of the point in the interior angle bisector of the angle . Prove that the lines 
, 
and 
are concurrent.
for all permutations 
of 
,
are homothetic, meaning that the lines connecting the corresponding points, namely 
,
all lie on the incircle.
.
.
concur at the centre of homothety.
and 
.
be the incenter of 
all lie on the incircle.
is the reflection of 
with respect to the 
so 
lies on the incircle. Similarly, 
and 
lie on the incircle. 
because of equal radii due to Claim 1.
,
.
But, 
so 
Thus, 
Similarly, 
Therefore, 
and we similarly get 
and 
be the midpoint of 
Then, 
is collinear.
Then, 
Thus, 
as desired
because a diameter is perpendicular to a chord at its midpoint and because 
But, 
so 
.
and 
so 
with corresponding parallel sides. Thus, there exists a homothety that maps 
to 
and 
and 
intersect at the center of the homothety. 
is equal to the distance from 
to 
which can be written as 
is homothetic to 
,
lies on the incircle.
