lying outside the circle 
,
.
.
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 
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 
.
.
Y by Davi-8191, tenplusten, HsuAn, Adventure10
Let 
be a fixed acute-angled triangle. Consider some points 
and 
lying on the sides 
and 
, respectively, and let 
be the midpoint of 
. Let the perpendicular bisector of intersect the line 
at 
, and let the perpendicular bisector of 
intersect the lines and at 
and 
, respectively. If the quadrilateral 
is cycle, prove that 
.
be a fixed acute-angled triangle. Consider some points 
and 
lying on the sides 
and 
, respectively, and let 
be the midpoint of 
. Let the perpendicular bisector of intersect the line 
at 
, and let the perpendicular bisector of 
intersect the lines and at 
and 
, respectively. If the quadrilateral 
is cycle, prove that 
.
then 
at its midpoint 
Since 
is the reflection of 
on 
then it follows by symmetry that 
cuts 
again at 
Thus again, by symmetry, if 
cuts 
then 
and 
are isogonals WRT 
are tangents of 
as 
,
specified in the problem statement). Now, let 
denote 
,
are reflections of 
,
,
are collinear. From 
we obtain 
From 
we find 
It follows that 
Now, because the reflection 
in 
,
(well-known). Meanwhile, since 
By Power of a Point, we obtain 
and 
Therefore, ![\[\frac{AX}{MX} \cdot \frac{MB}{AU} = \frac{AH}{MB} \cdot \frac{MV}{AV} = \frac{AH}{2MB} = \frac{OM}{MB},\]](http://latex.artofproblemsolving.com/f/1/b/f1b6a4f57e9f08c9f20ad8d2904c47b750824a98.png)
where 
Then by side-angle-side similarity, we deduce that 
,
and passes through 
on 
respectively. Let 
at ![\[
\angle FXE=\angle GXH=\angle TKS=180-\angle FAE
\]](http://latex.artofproblemsolving.com/7/f/1/7f1594c4446e957f0151b4e1140b1ebd033906da.png)
is cyclic. By Brokard's Theorem, 
,
to 
,
.
,
.
,
m
be two such points on the arc 
are concyclic, 
are also concyclic which is not possible as they lie on the same side of 
so that 
)
.
.
goes through midpoint of chord 
,
- 
is tangent to 
and result follows.
outside 
is cyclic
the circle with radius zero centered at 
we have that 
goes from the modpoints of 
so 
then 
is tangent t
.
so 
. It's 
is 
