AoPS Academy
be a triangle, and let 
be a point on side 
. A line through intersects side 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point 
distinct from point 
. The lines 
and 
intersect again at 
and 
respectively.
such that 
for all real number 
and 
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let intersect 
and 
at and respectively and intersect 
and 
at and respectively .
, point 
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds. is an isosceles triangle.
![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
be any triangle and let 
and 
be the symmetrics of the circumcenter wrt the three sides. Prove that the triangles and ![$[DEF]$](http://latex.artofproblemsolving.com/c/5/b/c5ba7afa12298272441c17f763cea35fd1402bf3.png)
are congruent.
is a parallelogram, therefore, using complex numbers and saying 
,
,
and 
is the image of 
,