Here, we begin with a quantity which is invariant under the group
of direct similarities, S^{+}(2). We derive the main invariants of
similarity geometry, those of ratio and size of angle.
Definition
If u,v,w,z are distinct points of the complex plane,
then the similarity ratio <u,v,w,z> is defined as
(uv)/(wz).
If t is an similarity transformation, then we may apply t to each
variable. We shall write t<u,v,w,z> for the similarity ratio of the
images, <t(u),t(v),t(w),t(z)>.
Theorem SI1
The similarity ratio is invariant under the group S^{+}(2).
Proof
Suppose that t is in S^{+}(2). Then t(z) = az+b, with a ≠ 0, so that
t(u)t(v) = (au+b)(av+b) = a(uv), and t(w)t(z) = a(wz).
In the quotient, the a's cancel, so we have the result.
An indirect similarity transformation t maps z to m(z)*, where
m is in S^{+}(2). From Theorem SI1, we can see that
t<u,v,w,z> 
= <m(u)*,m(v)*,m(w)*,m(z)*> 


= <m(u),m(v),m(w),m(z)>* 


= <u,v,w,z>* 
by Theorem SI1 
Convention We shall take the argument of a complex number
as the value in [π,π].
Theorem SI1*
If t is a similarity transformation, then it
(1) preserves the modulus of similarity ratios, and
(2) preserves or reverses the argument of similarity ratios.
Once we observe that the, if the points U,V,W,Z have complex
coordinates u,v,w,z, then the modulus of <u,v,w,z> is equal to
uv/wz = UV/WZ, the ratio of lengths. Thus we have
Theroem SI2
The ratio of lengths of segments is an invariant of similarity geometry.


To establish the invariance of angle, we consider first similarity
ratios of the form <u,z,v,z> i.e. (uz)/vz) for complex u,v and z.
Let U,V,Z be the points with complex coordinates u,v,z.
Then uz = r.exp(iα), where r = UZ, and α is as shown.
and vz = s.exp(iβ), where s = VZ, and β is as shown.
Thus, (uz)/(vz) = (r/s).exp(i(αβ), so that
the modulus of <u,z,v,z> is the ratio UZ/VZ, and
the argument of <u,z,v,z> is the signed angle VZU.
By Theorem SI1*, we know that the modulus is preserved, and
that the size of the argument (in [0,π]) is also preserved.
The ratio result is a special case of Theorem SI2. The other gives
Theorem SI3
The size of angles is an invariant of similarity geometry.

