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:
, whose vertices 
and lie on the sides 
and 
, respectively, of a given square 
. Find the locus of the midpoints of the sides 
of all such triangles .
be the midpoint of 
; see easily, 
are cyclic, and: 
, similarly 
, hence 
is equilateral, i.e. is a fix point; consequently the midpoint of , which is the circumcenter of 
lies on the perpendicular bisector of 
, done.
