.
Three complex numbers z1,z2,z3 are such that z1+z2+z3=0 and |z1|=|z2|=|z3|.prove that they represent the vertices
of an equilateral triangle in the complex plane.
~~~~~~~~~~~~~~~~~~~~~~~~~
There is one obvious solution z1 = z2 = z3 = 0.
But it is very degenerated solution, and we will put it aside and will not consider it more.
So, we will assume that |z1| =/= 0; i.e. |z1| > 0.
Then |z2| > 0 and |z3| > 0, too.
From this point, we start the TRUE solution.
Let us divide the equation
z1 + z2 + z3 = 0 (1)
by z1 (both sides). We can do it, because z1 =/= 0, according to the assumption. You will get
1 + z2/z1 + z3/z1 = 0 (2)
and |z2/z1| = |z3/z1| = 1 (since |z1|=|z2|=|z3| ).
Therefore, the original problem is equivalent to this modified formulation:
If z2 and z3 are complex numbers such that |z2| = |z3| = 1 and 1 + z2 + z3 = 0, then the points of the unit circle
(1,0), z2 and z3 are vertices of the equilateral triangle.
Thus we simplified the problem.
Now we can prove the modified formulation very simple.
Since the sum 1 + z2 + z3 is zero, its imaginary part is zero, too.
It means that imaginary parts of complex numbers
z2 and z3 have equal absolute values and opposite signs.
In other words, the points (the complex numbers) z2 and z3 lie in different sides of x-axis and are equally remoted from x-axis on the unit circle.
Hence, the segment connecting the points z2 and z3 in complex plane is perpendicular to x-axis.
Then the points z2 and z3 have equal x-coordinates.
In other words, their real parts are equal.
And since the sum of their real parts is equal to -1, the individual real parts are each.
So, the complex numbers z2 and z3 are
= and
= .
Together with the point (1,0) they constitute the vertices of the equilateral triangle.
Thus the modified statement is proved.
So, the equivalent to it the original statement is proved also.
Solved and completed.