Re: Complex numbers (for geometry proof)

On Fri, 16 Nov 2007 21:32:39 +0100, Denis Feldmann wrote:

Fons a écrit :
What would be a short and straightforward way to prove that

|z-z_1| / |z-z_2| = a

(in which z_1 and z_2 are complex and a is real and independent of z)
implies the existence of a complex number z_0 and a real number R (also
independent of z) so that

|z-z_0| = R

maybe without having to calculate z_0 and R?


Short and straightforward? i dont know. I would either write some
equations (and note that the resuting relation between real and
imaginary parts of z is that of a circkle), or use the geometrical
interpretation, getting what is known as "Leibniz problem"

Well, I can find the two points that are on the line connecting z_1 and
z_2 and those would form the diameter of the circle, allowing me to
calculate z_0 and R. But I was wondering if there was a nice and simple
way, perhaps doing only some operations on the complex numbers/
expressions themselves (e.g without explicitly going to the reals by
putting z=x+iy), to prove that the locus is "some" circle.





.

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