From 'Fields and Waves in Communications Electronics' by Ramo Whinnery and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is positive
so the reflected 'negative' current wave is moving left which gives the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)

then solving from (1) and (3) to get Vn/Vp

multiple (3) by Zload on both sides
Vp*Zload/Zo - Vn*Zload/Zo = Vload
substitute this for Vload in (1) to get:
Vp+Vn = Vp*Zload/Zo - Vn*Zload/Zo
group terms:
Vp-Vp*Zload/Zo = -Vn-Vn*Zload/Zo
factor:
Vp(1-Zload/Zo) = Vn(-1-Zload/Zo)
divide out terms
(1-Zload/Zo)/(-1-Zload/Zo) = Vn/Vp
multiply by Zo/Zo
(Zo-Zload)/(-Zo-Zload) = Vn/Vp
mulitply by -1/-1
(Zload-Zo)/(Zload+Zo) = Vn/Vp

therefo
rho = Vn/Vp = Zload-Zo/Zload+Zo

what could be simpler... apply kirchoff's and ohm's laws and a bit of
algebra.