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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 *k*A > *p*B but __it is
not the case__ that *k*C > *p*D.

We'll suppose *k*C < *p*D. [We leave the
case that** ***k*C = *p*D 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.*