AoPS Academy
be an acute-angled triangle with 
and let 
be the foot of the
-angle bisector on 
. The reflections of lines 
and 
in line meet and at points
and 
respectively. A line through meets and at 
and 
respectively such that and 
while lies strictly between 
and . Prove that the circumcircles of
and 
are tangent to each other.
with circumcircle 
. Point 
in such that 
is the angle bisector of 
. Circle with center and radius 
intersects and at and respectively, 
. Let 
be the midpoint of arc in that didn't have . Prove that 
angle bisector of 
if and only if 
.
be real numbers satisfying
+1 w
. Prove that:
be altitude, 
be median, 
be angle bisector of acute triangle . Denote 
the points of intersection of with and 
respectively. Find the perimeter of the triangle if 
and 
.
be altitude, be median, be angle bisector of acute triangle . Denote the points of intersection of with and respectively. Find the perimeter of the triangle if and .
So D is midpoint of BC.
.
to be midpoint of 
be G.
= 
= 
= 
and 
.
perimeter =
