be an acute triangle with 
,
be the projection from 
onto 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
sastifying 
Y by
Let 
and 
, respectively, be the point of intersection of the medians and the center of the circle circumscribed around 
The perpendicular bisectors of the segments 
, and 
intersect at points 
, and 
. Prove, that is the point of intersection of the medians of the triangle 
.
and 
, respectively, be the point of intersection of the medians and the center of the circle circumscribed around 
The perpendicular bisectors of the segments 
, and 
intersect at points 
, and 
. Prove, that is the point of intersection of the medians of the triangle 
.
