[/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 
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 
respectively. A line through 
and 
respectively such that 
while 
and 
and 
are tangent to each other.
with circumcircle 
.
.
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 agimog, Adventure10, Mango247
In an acute-angled and not isosceles triangle 
we draw the median 
and the height 
Points
and 
are marked on the lines 
and 
, respectively, so that the 
and 
.
The circumcircle of
intersects the line 
for second time at point 
Prove that 
M. Didin
we draw the median 
and the height 
Points
and 
are marked on the lines 
and 
, respectively, so that the 
and 
.The circumcircle of
intersects the line 
for second time at point 
Prove that 
M. Didin
passing through 
meets each other at 
.
be the isogonal conjugate of 
)
,
lie on a circle with diameter 
is the projection of 
.
is the projection of 
is the pedal circle of 
is the projection of 
due to 
share the same pedal circle (WRT 
.
is the midpoint of 
again at 
on 
of 
on 
But by Butterfly theorem for the cyclic 
it follows that 
is X-midline of 
or 
be the midpoint of 
and 
then applying butterfly theorem for the circle with diameter 
.
be the circumcenter of triangle 
is the center of 
,
hence 
.
or 
.
at 
or 
performing inversion wrt 
it amounts to showing that 
& 
are equidistant from 
with 
and 
as 'wings', it follows that 
(
.
as wings. 
in 
are on 
such that 
.
meet the lines through 
parallel to 
respectively at 
are concyclic.
to 
since we already know that 
are concyclic. Now, for this it suffices that 
.
and so 
.
and 
.
also has midpoint 
too.
.
be 
.
is a rectangle.
is reflection of 
, 
are collinear.
.
to 
as these circles are congruent and are reflections over 
,
,
and so 
.
and let 
at 
so 
is cyclic. Likewise, 
is cyclic. It follows that 
is the pedal circle of point 
,
,
;
be a parallelogram. By Reim's theorem, 
is tangent to 
and 
is tangent to 
.
,
.
is the midpoint of 
,
.
and 
be the reflections of 
and 
over 
and 
,
be the midpoint of 
.
,
,
,
is a Varignon parallelogram, and since 
,
.
be the reflection of 
.
.
are concyclic. Let 
be the orthic triangle of 
is the perpendicular bisector of 
,
.
,
Hence 
are concyclic, which means 
,
Since 
inversion wrt 
which fixes 
maps 
onto points 
respectively. These points are symmetric wrt 
,
.
meet 
.
we remark that 
is a rectangle.
and let 
denote the point on 
.
,
and 
are reflections of each other in 
.
.
in 
and thus 
be the reflection of 
be the reflection of 
.
.
,
and 
.
has equation 
by computing the slope of 
,
has equation 
,
-
.
has equation 
and 
,
,
-
.
,
is cyclic which finishes. 
and 
and 
and 
.
at 
.
respectively such that 
and 
.
lie on the same circle.
.
intersects line 
intersects 
at 
,
.
be the point symmetric to 
.
and 
.
and 
.
,
.
where 
