PROOF WITHOUT WORDS: $a, b, c, d>0$; $ \frac{a}{b}=\frac{c}{d} \Rightarrow \frac{a}{b}=\frac{c}{d}=\frac{s a+t c}{s b+t d} $

Abstract

A pictorial proof of the theorem, $\frac{a}{b}=\frac{c}{d} \Rightarrow \frac{a}{b}=\frac{c}{d}=\frac{s a+t c}{s b+t d}$, is presented for $a, b, c$, and $d>0$ and arbitrary real numbers $s$ and $t$; it is an extended version of the following theorem: $\frac{a}{b}=\frac{c}{d} \Rightarrow \frac{a}{b}=\frac{c}{d}$ $=\frac{a \pm c}{b \pm d}$. The diagrams used in this proof can also help college students understand the concept of linearity.

Received: June 26, 2024
Accepted: July 12, 2024