be an acute triangle with 
,
be the projection from 
.
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 
.
intersect 
.
.
sastifying 
Y by Adventure10, Mango247
Two circles 
and 
touch at 
. They have a common tangent which meets at 
and at 
. The points and are different. Let 
be a diameter of . Prove that 
and 
lie on a straight line.
See here: http://www.bmoc.maths.org/home/bmo1-2013.pdf
and 
touch at 
. They have a common tangent which meets at 
and at 
. The points and are different. Let 
be a diameter of . Prove that 
and 
lie on a straight line.See here: http://www.bmoc.maths.org/home/bmo1-2013.pdf
intersects the circle 
are parallel to 
.
at 
then as 
is parallel to 
so 
.
are collinear, so 
.
.
,
are collinear.
and 
are collinea
