If four straight lines are proportional then the rectangle contained by the (two) outermost is equal to the rectangle contained by the middle (two). And if the rectangle contained by the (two) outermost is equal to the rectangle contained by the middle (two) then the four straight lines will be proportional. * Let $AB$, $CD$, $E$, and $F$ be four proportional straight lines, (such that) as $AB$ (is) to $CD$, so $E$ (is) to $F$ * I say that the rectangle contained by $AB$ and $F$ is equal to the rectangle contained by $CD$ and $E$ (and conversely).
With $a:=|\overline{AB}|,$ $b:=|F|,$ $c:={|\overline{CD}|}$, and $d:=|E|,$ this proposition states that $\frac ac=\frac db$ if and only if $ab=cd.$
Proofs: 1