ELMO SL 2013 G3
by Wolstenholme, Aug 1, 2014, 9:31 PM
In 
, a point 
lies on line 
. The circumcircle of 
meets 
at 
(other than 
), and the circumcircle of 
meets 
at 
(other than ). Prove that as varies, the circumcircle of 
always passes through a fixed point other than , and that this point lies on the median from to .
Proof:
I will provide a barycentric coordinate solution.
Let
and let 
. Let 
for some real 
.
We find that the equation of the circumcircle of
is given by: 
and it is easy to see that this intersects line at the point 
. Similarly we find that 
.
Now it is easy to find that the circumcircle of
has equation 
. It is easy to compute that regardless of the value of , the point 
lies on this circumcircle, and moreover lies on the -median of 
so we are done.
The motivation for using barycentric coordinates is that despite the large number of circles in this diagram, all are defined by vertices or points on sides of and so their equations are easy to deal with. Moreover, the condition that our fixed point lies on the relevant median implies that its barycentric coordinates will be nice.
, a point 
lies on line 
. The circumcircle of 
meets 
at 
(other than 
), and the circumcircle of 
meets 
at 
(other than ). Prove that as varies, the circumcircle of 
always passes through a fixed point other than , and that this point lies on the median from to .Proof:
I will provide a barycentric coordinate solution.
Let
and let 
. Let 
for some real 
.We find that the equation of the circumcircle of
is given by: 
and it is easy to see that this intersects line at the point 
. Similarly we find that 
.Now it is easy to find that the circumcircle of
has equation 
. It is easy to compute that regardless of the value of , the point 
lies on this circumcircle, and moreover lies on the -median of 
so we are done.The motivation for using barycentric coordinates is that despite the large number of circles in this diagram, all are defined by vertices or points on sides of
and so their equations are easy to deal with. Moreover, the condition that our fixed point lies on the relevant median implies that its barycentric coordinates will be nice.
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