144. A conditioned concurrency in a triangle.
by Virgil Nicula, Oct 5, 2010, 3:24 PM
Let 
be a triangle. The its incircle 
touches the lines 
, 
in the points 
, 
. Denote the middlepoint 
of
the side![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Prove that the 
- simmedian and the lines 
, 
are concurrently if and only if 
.
Generalization I. Let be an acute triangle. The its incircle touches the lines , in the points , . Denote the middlepoint of
the side. Let ![$S\in [BC]$](//latex.artofproblemsolving.com/2/2/f/22f356e1f071074d7e4516232dd049d3a24034eb.png)
for which 
. Prove that the lines 
, , are concurrently if and only if 
.
Particular cases.
For 
, 
, i.e. 
obtain http://www.artofproblemsolving.com/Forum/viewtopic.php?t=141487
If the point 
is the foot of the -simmedian, i.e. 
, then 
.
Generalization II. Let be an acute triangle. The its incircle touches the lines , in the points , . Let 
be two points for which
and . Prove that the lines , , are concurrently if and only if 
.
Particular case. If
, i.e. the point is the middlepoint of the side , then obtain the generalization I.
be a triangle. The its incircle 
touches the lines 
, 
in the points 
, 
. Denote the middlepoint 
ofthe side
![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Prove that the 
- simmedian and the lines 
, 
are concurrently if and only if 
.Generalization I. Let be
an acute triangle. The its incircle touches the lines , in the points , . Denote the middlepoint ofthe side
. Let ![$S\in [BC]$](http://latex.artofproblemsolving.com/2/2/f/22f356e1f071074d7e4516232dd049d3a24034eb.png)
for which 
. Prove that the lines 
, , are concurrently if and only if 
.Particular cases.
For 
, 
, i.e. 
obtain http://www.artofproblemsolving.com/Forum/viewtopic.php?t=141487
If the point 
is the foot of the -simmedian, i.e. 
, then 
.Generalization II. Let be
an acute triangle. The its incircle touches the lines , in the points , . Let 
be two points for which
and . Prove that the lines , , are concurrently if and only if 
.Particular case. If
, i.e. the point is the middlepoint of the side , then obtain the generalization I.This post has been edited 6 times. Last edited by Virgil Nicula, Dec 1, 2015, 11:03 AM
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