"Brad Cooper" <Brad.Cooper_17@bigpond.com> wrote in message news:WL8gq.3088$NR2.firstname.lastname@example.org...
> 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. > > 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.
Let p_i = 1/z_i, i=1,2,3 .
Now, p_1 = (p_2+p_3)/2, so p_1, p_2, p_3 are collinear. Suppose they lie on the line L.
Inverting this figure wrt the unit circle centred at 0, and remembering that the inverse of any complex number wrt that circle is simply the complex conjugate of its reciprocal, we see that the line L inverts into a circle passing through the origin, on which the conjugates of the points z_1, z_2 and z_3 lie. Hence, z_1, z_2, and z_3 themselves lie on a circle passing through the origin.