
Re: Complex Numbers  Argand Diagram Question
Posted:
Sep 27, 2011 12:47 PM


On Tue, 27 Sep 2011 10:27:32 +1000, "Brad Cooper" <Brad.Cooper_17@bigpond.com> wrote:
>Complex Numbers  Argand Diagram Question > >In Gow's book, A Course in Pure Mathematics, I have managed to answer all of >the complex number questions in Chapter 6 except question 21. > > > >21. If the complex numbers z_1, z_2, z_3 are connected by the relation > >2/z_1 = 1/z_2 + 1/z_3 > >show that the points Z1, Z2, Z3 representing them in an Argand diagram lie >on a circle passing through the origin.
This business of "Argand diagrams" is a bit oldfashioned; in fact, at least the way mathematicians look at it these days, a complex number is exactly the same thing as the point "representing it in an Argand diagram".
That is, Z_j = z_j; below I'm just going to talk about z_j.
Anyway, how you prove this depends on what you know. Say w_j = 1/z_j. Then
w_1 = (w_2 + w_3)/2,
which says that w_1 is the midpoint of the segment joining z_2 and z_3. In particular, there is a straight line L such that w_1, w_2 and w_3 all lie on L.
So all you have to do is show that if L is a line in the plane then C = {1/w : w in L} is a circle in the plane passing through the origin, minus the origin...
>Note: _1, _2, _3 denote subscripts. > > > >I have tried algebraic and geometrical approaches. One attempt was to show >that the three points, along with the origin, form a cyclic quadrilateral. > >I used a CAS to draw a diagram for the problem, but after many hours I don't >seem to be any closer to finding a solution. > >Any help much appreciated. > >Cheers, >Brad > >

