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, 
, 
define 
to be the semiperimeter of 
. Let 
be a triangle and let 
be on side 
such that![\[\frac{s(ABD)}{BD}=\frac{s(ACD)}{CD}\]](http://latex.artofproblemsolving.com/9/4/f/94f0bcb2cf847452d6e6197d5bb9b6f0ffd72f6d.png)
Let 
be a point on 
. Let 
and 
be on 
and 
such that 
and 
. Prove that there exists a line 
parallel to such that the circumcircles of 
and 
intersect at two fixed points.
+1 w
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 
.
is a triangle with 
its circumcircle, 
the midpoint of the minor arc 
of the circle 
and 
a point on such that 
is perpendicular to . If 
are the intouch points of the incircle of with the sides 
, prove that the lines 
are concurrent.
is a triangle with its circumcircle, the midpoint of the minor arc of the circle and a point on such that is perpendicular to . If are the intouch points of the incircle of with the sides , prove that the lines are concurrent.
such that:
.
,
which implies the conclusion.