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.
, see that 
and, with 
we are done.
be a trapezoid with 
.
and 
are constructed. Prove that lines 
,
and 
are concurrent.
the intersection point of 
,
,
and 
.
is homothetic to 
.
is homothetic to 
- that is why 
,
are concurrent.
be the intersection of 
and 
.
is similar to 
,
is similar to 
.
and 
.
.
and 
are similar. The spiral similarity which brings 
to 
.
to 
,
.
are collinear. So