53. An remarkable concurrency.
by Virgil Nicula, Jul 9, 2010, 10:15 PM
Quote:
Problem. Given a triangle 
and the midpoints 
, 
, 
of its sides ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, ![$[CA]$](//latex.artofproblemsolving.com/4/5/c/45c1acd47628de406680d04c09fe6314c3847acf.png)
, ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. Let 
be a point in the plane of triangle and let the lines 
, 
, 
intersect the lines 
, 
, 
at the points 
, 
, 
respectively. Denote by 
, 
, 
the midpoints of the segments 
, 
, 
. Prove that the lines 
, 
and 
concur at one point.
and the midpoints 
, 
, 
of its sides ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, ![$[CA]$](http://latex.artofproblemsolving.com/4/5/c/45c1acd47628de406680d04c09fe6314c3847acf.png)
, ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. Let 
be a point in the plane of triangle and let the lines 
, 
, 
intersect the lines 
, 
, 
at the points 
, 
, 
respectively. Denote by 
, 
, 
the midpoints of the segments 
, 
, 
. Prove that the lines 
, 
and 
concur at one point.Proof. Since
and are the midpoints of and ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
of 
, we must have 
. In other words, 
. Similarly, 
. Thus, , and lie on oneand the same line parallel to
. In other words, lies on 
and we have 
. Hence, by the Thales' theorem, 
. Similarly, lies on 
and satisfies
and lies on 
and satisfies 
. Thus, 
. But , , concur (namely, at ).Thus, by Ceva's theorem,
. Hence, 
. Thus, by Ceva's theorem, applied to 
and to 
, 
, 
obtainthat
, and must concur.Virgil Nicula wrote:
Let be a triangle and the points 
with barycentric coodonates relative to the triangle .
I note
which belong to 
respectively such that 
.
Prove that is the centroid of 
iff there is the relations 
.
be a triangle and the points 
with barycentric coodonates relative to the triangle .I note
which belong to 
respectively such that 
.Prove that
is the centroid of 
iff there is the relations 
.This post has been edited 6 times. Last edited by Virgil Nicula, Nov 27, 2015, 7:27 AM
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