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be an acute-angled triangle with 
, orthocenter 
, circumcircle 
and circumcentre 
. Let 
be the midpoint of 
and let 
be a point such that 
is a parallellogram. Suppose that there exists a point 
on and on the opposite side of 
to 
such that 
. Let 
be the midpoint of 
. Prove that if 
, then 
.
be an acute scalene triangle , be the orthogonal projection of on , and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
be a triangle and the points 
and 
on , and on , and and on be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are and 
, respectively. Prove that the lines 
and 
are concurrent.
. Then 
is cyclic and 
.
. Since 
and 
we have that 
is the diameter of the circumscribed circle of 
. Then 
.
is cyclic, which implies that, 
Also, 
, Now since, 
be a square and 
points on sides 
respectively such that 
.
where 
are points of intersection of 
.
it's said 
?
and 
are cyclic. angle bash some more. and then prove 
is cyclic, upon noticing 
is diameter. 
is centre hence 
oops sorry. i thought ur soln had an error. but it was actually a typo. sorry 
