83. A metrical characterization of concurrence.
by Virgil Nicula, Aug 7, 2010, 9:27 PM
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=360757
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=38&t=367293
Let
with area ![$[ABC]=1$](//latex.artofproblemsolving.com/4/a/0/4a08c4c1431ba612a7f5f04de67b275518a068b2.png)
and lines 
, 
. Denote some intersections of given lines with sidelines of :
![$\boxed{\blacktriangleright\left|\begin{array}{c}
X\in d_2\cap d_3\\\
Y\in d_3\cap d_1\\\
Z\in d_1\cap d_2\end{array}\right|\implies [XYZ]=\frac {1}{\lambda}\cdot\Delta^2=\frac {1}{\lambda}\cdot \left|\begin{array}{ccc}
1 & m & n\\\
q & 1 & p\\\
r & s & 1\end{array}\right|^2}$](//latex.artofproblemsolving.com/f/8/9/f89c6dd94b9c26539003f91c75db6825b375b741.png)
where
.
Proof 1 (without the second point of conclusion).[/size] Choose an origin
of the vectorial system and for any point 
denote 
. Thus,
. Define 
, 
and 
.
Observe that 
. Apply Menelaus' theorem to 
. Apply Menelaus' theorem in 
.
Observe that 
. Apply Menelaus' theorem in 
.
Apply Menelaus' theorem in
.
Thus,
.
Proof 2 (with the second point of conclusion).
equation of 
is 
, i.e. 
equation of 
is 
, i.e. 
equation of 
is 
, i.e. 
.
In conclusion, 
and
![$[XYZ]=\frac {1}{\lambda}\cdot \left|\begin{array}{ccc}
1-ps & pr-q & qs-r\\\\
ns-m & 1-nr & mr-s\\\\
mp-n & nq-p & 1-mq\end{array}\right|\stackrel{(*)}{=}\frac {1}{\lambda}\left|\begin{array}{ccc}
1 & m & n\\\\
q & 1 & p\\\\
r & s & 1\end{array}\right|^2$](//latex.artofproblemsolving.com/8/b/8/8b8fe6daada3ae373796b6c4ef00ec1dff376a4c.png)
where .
Indeed, observe that 
.
Remark.
.
Two remarkable particular cases.
and 
i.e. 
and 
.
and 
i.e. and 
Apply the Menelaus' theorem to
and 
: 
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=38&t=367293
Let
with area ![$[ABC]=1$](http://latex.artofproblemsolving.com/4/a/0/4a08c4c1431ba612a7f5f04de67b275518a068b2.png)
and lines 
, 
. Denote some intersections of given lines with sidelines of :
![$\boxed{\blacktriangleright\left|\begin{array}{c}
X\in d_2\cap d_3\\\
Y\in d_3\cap d_1\\\
Z\in d_1\cap d_2\end{array}\right|\implies [XYZ]=\frac {1}{\lambda}\cdot\Delta^2=\frac {1}{\lambda}\cdot \left|\begin{array}{ccc}
1 & m & n\\\
q & 1 & p\\\
r & s & 1\end{array}\right|^2}$](http://latex.artofproblemsolving.com/f/8/9/f89c6dd94b9c26539003f91c75db6825b375b741.png)
where
.Proof 1 (without the second point of conclusion).[/size] Choose an origin
of the vectorial system and for any point 
denote 
. Thus,
. Define 
, 
and 
.
Observe that 
. Apply Menelaus' theorem to 
. Apply Menelaus' theorem in 
.
Observe that 
. Apply Menelaus' theorem in 
.Apply Menelaus' theorem in
.Thus,
.Proof 2 (with the second point of conclusion).
equation of 
is 
, i.e. 
equation of 
is 
, i.e. 
equation of 
is 
, i.e. 
.In conclusion,
and![$[XYZ]=\frac {1}{\lambda}\cdot \left|\begin{array}{ccc}
1-ps & pr-q & qs-r\\\\
ns-m & 1-nr & mr-s\\\\
mp-n & nq-p & 1-mq\end{array}\right|\stackrel{(*)}{=}\frac {1}{\lambda}\left|\begin{array}{ccc}
1 & m & n\\\\
q & 1 & p\\\\
r & s & 1\end{array}\right|^2$](http://latex.artofproblemsolving.com/8/b/8/8b8fe6daada3ae373796b6c4ef00ec1dff376a4c.png)
where
.
Indeed, observe that 
.Remark.
.Quote:
Let 
, 
, 
be the middlepoints of the sides in the triangle 
For define 
, 
, 
Prove that the lines
, 
, 
are concurrently if and only if 
See http://www.artofproblemsolving.com/Forum/viewtopic.php?t=114378
, 
, 
be the middlepoints of the sides in the triangle 
For define 
, 
, 
Prove that the lines
, 
, 
are concurrently if and only if 
See http://www.artofproblemsolving.com/Forum/viewtopic.php?t=114378 Two remarkable particular cases.
and 
i.e. 
and 
. 
and 
i.e. and 
Apply the Menelaus' theorem to
and 
: 
This post has been edited 60 times. Last edited by Virgil Nicula, Nov 23, 2015, 3:23 PM
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