
Specific Constructions 97
If z,w are collinear with 0, circle C[z, |w|] meets L[0,z] in two points, z + w and
z −w.
(2) The circle C[0,|z|] meets the line L[0, z] at z and at −z.
(3) Consider the triangle T with vertices 0,1,z. Construct point p so that the triangle
with vertices 0,w, p is similar to T .
We claim that p = zw. By similarity |p|/|w| = |z|/1, so |p| = |z||w|. Further,
arg(p) = arg z + argw, where arg denotes the argument. Therefore p = zw.
(4) Let C[0,1] meet L[0, z] at p (with 0 lying between z and p). Then |p| = 1.
Construct a triangle with vertices 0, p,q similar to 0,z,1. Then |q|/1 = |p|/|z| =
1/|z|, so |q = 1/|z|.
Let C[0,q] meet L