130. A triangle and a fixed point.
by Virgil Nicula, Sep 24, 2010, 1:09 PM
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=368627
Proof. Suppose w.l.o.g. that 
. Observe that 
, 
, i.e. 
, 
. Denote 
.
Apply the Menelaus' theorem to the transversals :
, i.e. the point 
is fixed and it is the intersection of the exterior 
-angle bisector of 
with
the sideline
. Observe that 
, 
.
Since
and 
obtain 
, i.e.
the tangent line in to the circumcircle of 
meet the sideline in the same point .
Quote:
Let 
be a triangle . Denote the midpoint 
of the side ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
and construct the points 
, 
so that 
. For a mobile point
define 
, 
. Prove that the line 
pass through a fixed point 
for which 
and 
,
i.e. the sideline , the exterior -angle bisector of and the tangent line in to the circumcircle of are concurrently !
be a triangle . Denote the midpoint 
of the side ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
and construct the points 
, 
so that 
. For a mobile point
define 
, 
. Prove that the line 
pass through a fixed point 
for which 
and 
,i.e. the sideline
, the exterior 
-angle bisector of 
and the tangent line in to the circumcircle of 
are concurrently !
. Observe that 
, 
, i.e. 
, 
. Denote 
.Apply the Menelaus' theorem to the transversals :
, i.e. the point 
is fixed and it is the intersection of the exterior -angle bisector of withthe sideline
. Observe that 
, 
.Since
and 
obtain 
, i.e.the tangent line in
to the circumcircle of meet the sideline in the same point .This post has been edited 12 times. Last edited by Virgil Nicula, Nov 23, 2015, 7:27 AM
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