Arithmetic proofs

You can see the arithmetic proof for the shortest diagonal and the side in the case of the following polygons:


Regular pentagon

Regular hexagon

Note that to say that two magnitudes \(d\) and \(l\) are commensurable, that is, there exists a measure common to both, is the same as saying that their ratio is a rational number. Indeed, if \(x\) is a measure common to \(d\) and \(l\), then \(d\) and \(l\) are both integer multiples of \(x\), that is, there exist two integer numbers \(m\) and \(n\) such that \(d=mx\) and \(l=nx\), so \(\frac{d}{l}=\frac{mx}{nx}=\frac{m}{n}\) and \(\frac{d}{l}\) is a rational number. Conversely, if \(\frac{d}{l}=\frac{m}{n}\), with \(m\) and \(n\) positive integers, then \(\frac{1}{m}d=\frac{1}{m}\frac{d}{l}l=\frac{1}{m}\frac{m}{n}l=\frac{1}{n}l\) and, taking \(x=\frac{1}{m}d=\frac{1}{n}l\), we have \(d=mx\) and \(l=nx\), so \(x\) is a measure common to \(d\) and \(l\).