[/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 
.
and let 
be the foot of the
.
and 
in line 
and 
respectively. A line through 
and 
respectively such that 
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 
.
.
Y by Adventure10, Mango247
In the plane, consider a circle with center 
and radius 
Let 
be an arbitrary triangle having this circle as its incircle, and assume that 
Find the locus of
a.) all vertices
of such triangles;
b.) all vertices
of such triangles;
c.) all vertices
of such triangles.
and radius 
Let 
be an arbitrary triangle having this circle as its incircle, and assume that 
Find the locus ofa.) all vertices
of such triangles;b.) all vertices
of such triangles;c.) all vertices
of such triangles.
inscribed in 
,
are the points where 
touch 
be the midpoints of 
respectively. We have 
,
implies 
.
will be the images of those (respectively) through the inversion wrt 
.
with 
taking any value in the interval ![$(0,\frac \pi 3]$](http://latex.artofproblemsolving.com/3/d/c/3dc09cb6ee4647e2cf7727d293e4413393cd2015.png)
,
,
,
.
,
.
,
.
is the middle angle, and we have to find bounds for it. It can take values as close to 
as we want (we take isosceles triangles s.t. 
one of the other angles), but it must be 
,
.
as we want, so the locus of 
,
.
