USA TST 2014 #1
by Wolstenholme, Oct 15, 2014, 4:26 AM
Let 
be an acute triangle, and let 
be a variable interior point on the minor arc 
of its circumcircle. Let 
and 
be the feet of the perpendiculars from to lines 
and 
, respectively. Let 
be the intersection of line 
and the perpendicular from 
to 
. Let 
be the line through parallel to 
. Prove that as varies along minor arc , the line always passes through a fixed point. (Specifically: prove that there is a point 
, determined by triangle , such that no matter where is on arc , line passes through .)
Proof:
We proceed with complex numbers. Let
be the orthocenter of 
and let 
have complex coordinates 
respectively. WLOG assume that the circumcircle of is the unit circle.
It is clear that
and that 
. Therefore line 
has equation 
. Since 
we find that line 
has equation 
.
Computing the coordinates of
, the intersection of these two lines, we find that 
. But since 
this implies that 
so quadrilateral 
is a parallelogram and so 
, hence, is the desired fixed point.
be an acute triangle, and let 
be a variable interior point on the minor arc 
of its circumcircle. Let 
and 
be the feet of the perpendiculars from to lines 
and 
, respectively. Let 
be the intersection of line 
and the perpendicular from 
to 
. Let 
be the line through parallel to 
. Prove that as varies along minor arc , the line always passes through a fixed point. (Specifically: prove that there is a point 
, determined by triangle , such that no matter where is on arc , line passes through .)Proof:
We proceed with complex numbers. Let
be the orthocenter of 
and let 
have complex coordinates 
respectively. WLOG assume that the circumcircle of is the unit circle.It is clear that
and that 
. Therefore line 
has equation 
. Since 
we find that line 
has equation 
.Computing the coordinates of
, the intersection of these two lines, we find that 
. But since 
this implies that 
so quadrilateral 
is a parallelogram and so 
, hence, is the desired fixed point.
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