Notez cu x lungimea laturii patratului.
[tex]\Delta NAP\sim\Delta NDC\Rightarrow \dfrac{n}{x+n}=\dfrac{AP}{x}\Rightarrow AP=\dfrac{nx}{x+n}[/tex]
[tex]\Delta MBP\sim\Delta MCD\Rightarrow \dfrac{m}{x+m}=\dfrac{PB}{x}\Rightarrow PB=\dfrac{mx}{x+m}[/tex]
[tex]AP+PB=x\Rightarrow\dfrac{nx}{x+n}+\dfrac{mx}{x+m}=x[/tex] Simplificam prin x, apoi aducem la acelasi numitor;
[tex]n(m+x)+m(x+n)=(x+n)(x+m)[/tex]
[tex]mn+nx+mx+mn=x^2+nx+mx+mn[/tex]
[tex]x^2=mn\Rightarrow x=\sqrt{mn}[/tex]
