56. A extremum problem with areas.
by Virgil Nicula, Jul 10, 2010, 1:42 AM
PP. Let 
be any line passing through the centroid 
of a triangle 
. Prove that the area of 2 parts of (divided by ) differ by not more than ![$1/9\cdot [ABC]$](//latex.artofproblemsolving.com/2/e/d/2ed41756eb07541d63e58132c33e8e2aede67268.png)
.
Proof. I can suppose
, 
. Denote 
, 
, where ![$x,y\in {\left[ \frac 12 ,1\right]},\ [MAN]=xy\cdot S$](//latex.artofproblemsolving.com/7/0/e/70e7c1a93fa93717e4f2e1639a38504f5eaabde4.png)
. Is wellknown the relation 
, i.e.
. Consequently, ![$[MAN]-\min$](//latex.artofproblemsolving.com/9/b/d/9bde65fca16b1b85276bbe60da78b752bc313381.png)
, where (constant) 
. Thus, ![$\overline{\underline{\left| [MAN]\ge \frac 49 \cdot S\right| }}\ \ (1)$](//latex.artofproblemsolving.com/7/9/3/793898efdbbd1cde4caeda5837cc8da41e7f0d58.png)
.
I note
, 
![$u,v\in [1,2]$](//latex.artofproblemsolving.com/6/a/d/6ad9f0dbdda902cdbdd8567bb8185add0b13fbbd.png)
, 
. Consequently, ![$[MAN]-\max$](//latex.artofproblemsolving.com/8/6/f/86ffaeff8e271497936740069fb371d7b7e33a23.png)
, where ![$u\in [1,2]$](//latex.artofproblemsolving.com/9/6/4/964e944a4c6054536a0047a2b59daf42e7bf1737.png)
.
Thus,![$\overline{\underline{\left| [MAN]\le \frac 12 \cdot S\right| }}\ \ (2)$](//latex.artofproblemsolving.com/f/9/7/f979f7659e87d4a11aa738567131ffb5079d96d5.png)
. In conclusion, from the relations 
and 
results that ![$|[BCNM]-[MAN]|=|S-2\cdot [MAN]|\in \left[ 0,\frac 19\right]$](//latex.artofproblemsolving.com/0/a/3/0a353a30dab916f8c813bca6316b345c6424a9e2.png)
.
Remark. If
, Then 
. You can propose a alike problem !
Generalization. Let
, a point 
, 
, 
(barycentric coordinates) and
, 
such that 
. Prove that ![$\overline{\underline{\left| 4yz\cdot S\le [MAN]\le \frac{\max\ \{ y,z \} }{x+\max\ \{ y,z \} }\cdot S\right|}}\ \ \ (*)$](//latex.artofproblemsolving.com/0/2/7/027120b6412dfb7ed1e37180cd423c695827e5b9.png)
.
Remarks. The point![$\ P\in [ABC]$](//latex.artofproblemsolving.com/4/2/5/42564c446f4314812760c0c56eb86038225a37a8.png)
, because . In the relation (*) we have equalities iff 
.
Some particular cases
1.
2.
. (For 
, the points M,N belong to the circle which is tangent to the lines AB,AC and to the circumcircle).
be any line passing through the centroid 
of a triangle 
. Prove that the area of 2 parts of (divided by ) differ by not more than ![$1/9\cdot [ABC]$](http://latex.artofproblemsolving.com/2/e/d/2ed41756eb07541d63e58132c33e8e2aede67268.png)
.Proof. I can suppose
, 
. Denote 
, 
, where ![$x,y\in {\left[ \frac 12 ,1\right]},\ [MAN]=xy\cdot S$](http://latex.artofproblemsolving.com/7/0/e/70e7c1a93fa93717e4f2e1639a38504f5eaabde4.png)
. Is wellknown the relation 
, i.e.
. Consequently, ![$[MAN]-\min$](http://latex.artofproblemsolving.com/9/b/d/9bde65fca16b1b85276bbe60da78b752bc313381.png)
, where (constant) 
. Thus, ![$\overline{\underline{\left| [MAN]\ge \frac 49 \cdot S\right| }}\ \ (1)$](http://latex.artofproblemsolving.com/7/9/3/793898efdbbd1cde4caeda5837cc8da41e7f0d58.png)
.I note
, 
![$u,v\in [1,2]$](http://latex.artofproblemsolving.com/6/a/d/6ad9f0dbdda902cdbdd8567bb8185add0b13fbbd.png)
, 
. Consequently, ![$[MAN]-\max$](http://latex.artofproblemsolving.com/8/6/f/86ffaeff8e271497936740069fb371d7b7e33a23.png)
, where ![$u\in [1,2]$](http://latex.artofproblemsolving.com/9/6/4/964e944a4c6054536a0047a2b59daf42e7bf1737.png)
.Thus,
![$\overline{\underline{\left| [MAN]\le \frac 12 \cdot S\right| }}\ \ (2)$](http://latex.artofproblemsolving.com/f/9/7/f979f7659e87d4a11aa738567131ffb5079d96d5.png)
. In conclusion, from the relations 
and 
results that ![$|[BCNM]-[MAN]|=|S-2\cdot [MAN]|\in \left[ 0,\frac 19\right]$](http://latex.artofproblemsolving.com/0/a/3/0a353a30dab916f8c813bca6316b345c6424a9e2.png)
.Remark. If
, Then 
. You can propose a alike problem !Generalization. Let
, a point 
, 
, 
(barycentric coordinates) and
, 
such that 
. Prove that ![$\overline{\underline{\left| 4yz\cdot S\le [MAN]\le \frac{\max\ \{ y,z \} }{x+\max\ \{ y,z \} }\cdot S\right|}}\ \ \ (*)$](http://latex.artofproblemsolving.com/0/2/7/027120b6412dfb7ed1e37180cd423c695827e5b9.png)
.Remarks. The point
![$\ P\in [ABC]$](http://latex.artofproblemsolving.com/4/2/5/42564c446f4314812760c0c56eb86038225a37a8.png)
, because . In the relation (*) we have equalities iff 
.Some particular cases
1.
2.
. (For 
, the points M,N belong to the circle which is tangent to the lines AB,AC and to the circumcircle).This post has been edited 5 times. Last edited by Virgil Nicula, Apr 6, 2016, 1:02 PM
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