IMO 2010 #4
by Wolstenholme, Nov 4, 2014, 3:42 PM
Let 
be a point interior to triangle 
(with 
). The lines 
, 
and 
meet again its circumcircle 
at 
, 
, respectively 
. The tangent line at 
to meets the line 
at 
. Show that from 
follows 
.
Proof:
Since
and 
we have that 
and 
By diving these two equations we obtain 
so it suffices to show that this product equals 
Now since 
we have that 
is tangent to the circumcircle of 
Therefore we have that 
so 
Similarly since 
is tangent to 
we have that 
so 
Combining this information we obtain
as desired.
be a point interior to triangle 
(with 
). The lines 
, 
and 
meet again its circumcircle 
at 
, 
, respectively 
. The tangent line at 
to meets the line 
at 
. Show that from 
follows 
.Proof:
Since
and 
we have that 
and 
By diving these two equations we obtain 
so it suffices to show that this product equals 
Now since 
we have that 
is tangent to the circumcircle of 
Therefore we have that 
so 
Similarly since 
is tangent to 
we have that 
so 
Combining this information we obtain
as desired.
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