183. A remarkable characterization of equilateral triangle.
by Virgil Nicula, Dec 4, 2010, 9:59 AM
A remarkable property. Let 
be a triangle , where 
, 
, 
, I denoted 
- the point 
from the
complex plane with the affix
. Then the triangle is equilateral iff 
.
Proof 1. is equilateral 
.
Proof 2. Let
. Thus, 
. Therefore, is equilateral
because 
.
Proof 3. Denote the midpoint
of the side ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Therefore, is equilateral triangle 
, where 
![$\left[\sqrt 3\cdot (b-c)\right]^2=(b+c-2a)^2$](//latex.artofproblemsolving.com/c/9/d/c9d14a2f2bd0b02862bb10c121d005843114ec33.png)
.
Remark. Prove easily that
(movement of translation). Therefore I can suppose w.l.o.g.
that
and 
, i.e. in the equilateral 
we have centroid 
, where 
is the circumcircle of .
In this case
and 
, i.e, 
.
be a triangle , where 
, 
, 
, I denoted 
- the point 
from thecomplex plane with the affix
. Then the triangle is equilateral iff 
. Proof 1.
is equilateral 
.Proof 2. Let
. Thus, 
. Therefore, is equilateral 
because 
.Proof 3. Denote the midpoint
of the side ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Therefore, is equilateral triangle 
, where 
![$\left[\sqrt 3\cdot (b-c)\right]^2=(b+c-2a)^2$](http://latex.artofproblemsolving.com/c/9/d/c9d14a2f2bd0b02862bb10c121d005843114ec33.png)
. Remark. Prove easily that
(movement of translation). Therefore I can suppose w.l.o.g.that
and 
, i.e. in the equilateral 
we have centroid 
, where 
is the circumcircle of .In this case
and 
, i.e, 
.This post has been edited 16 times. Last edited by Virgil Nicula, Nov 22, 2015, 6:28 PM
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