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Topic: Complex Numbers - Argand Diagram Question
Replies: 3   Last Post: Sep 27, 2011 12:48 PM

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 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Complex Numbers - Argand Diagram Question
Posted: Sep 27, 2011 12:47 PM

On Tue, 27 Sep 2011 10:27:32 +1000, "Brad Cooper"

>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 old-fashioned; 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,
>
>

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