Euclidean Proportion and Measurements

Proposition: For segments A,B,C,and D with m(A)=a,m(B)=b, m(C)=c, and m(D)=d:

1. If A:B::C:D then a/b=c/d .
2. If a/b=c/d then A:B::C:D

Proof:
1. We suppose a/b < c/d and show that the proportion must fail to be true.
Assume a/b < c/d . Then there must be some rational number p/k where a/b <p/k< c/d . [ This is true because between any two distinct real numbers there is a rational number. ] Then ka<pb and pd< kc. So kA<pB while pD<kC which shows that the proportion A:B::C:D is not true.

2. We suppose A:B is not proportional to C:D and we will find a contradiction of the hypothesis a/b=c/d.
Assume A:B is not proportional to C:D, so we can assume there are numbers k and p so that kA > pB but it is not the case that kC > pD.
We'll suppose kC < pD. [We leave the case that kC = pD as an exercise for the reader.]
Using the corresponding measurements we have: ka > pb but kc < pd.
Since these are all positive numbers we have by multiplication that kp ad > kp bc which contradicts ad=bc, an immmediate consequence of the hypothesis that a/b=c/d.