IMO 2009 #2
by Wolstenholme, Nov 5, 2014, 9:36 PM
Let 
be a triangle with circumcentre 
. The points 
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
. respectively, and let 
be the circle passing through and . Suppose that the line is tangent to the circle . Prove that 
Proof:
Let
denote the circumcircle of 
Now it is clear that 
and 
so we have that 
Moreover since 
is tangent to 
we have that 
so 
This implies that 
(since 
and 
) so 
and 
have the same power with respect to which implies that 
as desired.
be a triangle with circumcentre 
. The points 
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
. respectively, and let 
be the circle passing through and . Suppose that the line is tangent to the circle . Prove that 
Proof:
Let
denote the circumcircle of 
Now it is clear that 
and 
so we have that 
Moreover since 
is tangent to 
we have that 
so 
This implies that 
(since 
and 
) so 
and 
have the same power with respect to which implies that 
as desired.
August 2016
June 2016