123. My extension and its proof.
by Virgil Nicula, Sep 13, 2010, 3:56 AM
PP. Let the incircle 
of 
which touches 
, 
, 
at the points 
, 
, 
. The parallels to 
, 
,
through 
meet the lines , , at the points 
, 
, 
respectively. Prove that is the centroid of 
.
Proof (vectorial). Denote
and 
, where 
is the origin of vectorial system. Thus, 
, where 
From
![$2s\cdot D=[(s-c)\cdot B+(s-b)\cdot C]+[b\cdot B+c\cdot C]$](//latex.artofproblemsolving.com/e/1/2/e12545b90bc8c0febe26bb5a4edf9a148b19d7a0.png)
. Obtain so 
is the centroid of .
Generalization. Let and 
, 
with normalized barycentric co-ordinates w.r.t. . Let , , which belong to , , respectively
so that
, 
, 
. Prove that 
is the centroid of 
exist relations 
.
Particular case. . If
is the incenter and 
is the Gergogne's point, then obtain the propose problem.
Proof. Denote , where is origin of vectorial system, 
, 
and another two pairs of analogous 
. Suppose
w.l.o.g. that
and 
. From relations 
and 
obtain 
. Obtain analogous , . Thus, 
and is the centroid of 
. Exists
, 
, 
so that 
. Relations 
becomes 
and 
. In my opinion this problem is a nice extension with a
nice proof and mostly with a nice conclusion !
Proposed problem. The incircle of touches , , at the points , , . Let 
be the Nagel's point of .
Denote the points
, where . Prove easily that exists 
. The parallels
lines to
, 
, 
through meet the lines , , at the points , , respectively. Prove that is the centroid of 
.
Remark. Can construct the point
so : 
-Nagel cevian meets in 
; find the point 
such that 
.
of 
which touches 
, 
, 
at the points 
, 
, 
. The parallels to 
, 
,
through 
meet the lines , , at the points 
, 
, 
respectively. Prove that is the centroid of 
.Proof (vectorial). Denote
and 
, where 
is the origin of vectorial system. Thus, 
, where 
From
![$2s\cdot D=[(s-c)\cdot B+(s-b)\cdot C]+[b\cdot B+c\cdot C]$](http://latex.artofproblemsolving.com/e/1/2/e12545b90bc8c0febe26bb5a4edf9a148b19d7a0.png)
. Obtain so 
is the centroid of .Generalization. Let
and 
, 
with normalized barycentric co-ordinates w.r.t. . Let , , which belong to , , respectivelyso that
, 
, 
. Prove that 
is the centroid of 
exist relations 
.Particular case. . If
is the incenter and 
is the Gergogne's point, then obtain the propose problem.Proof. Denote
, where is origin of vectorial system, 
, 
and another two pairs of analogous 
. Supposew.l.o.g. that
and 
. From relations 
and 
obtain 
. Obtain analogous , . Thus, 
and
is the centroid of 
. Exists
, 
, 
so that 
. Relations 
becomes 
and 
. In my opinion this problem is a nice extension with anice proof and mostly with a nice conclusion !
Proposed problem. The incircle
of touches , , at the points , , . Let 
be the Nagel's point of .Denote the points
, where . Prove easily that exists 
. The parallelslines to
, 
, 
through meet the lines , , at the points , , respectively. Prove that is the centroid of 
.Remark. Can construct the point
so : 
-Nagel cevian meets in 
; find the point 
such that 
. This post has been edited 56 times. Last edited by Virgil Nicula, Nov 26, 2015, 9:30 PM
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