[tex]Notam:\\\\
~x~- ~lungimea ~laturii ~patratului.\\\\
\· NAP\sim\· NDC\Rightarrow \dfrac{n}{x+n}=\dfrac{AP}{x}\Rightarrow AP=\dfrac{nx}{x+n}\\\\
\· MBP\sim\· MCD\Rightarrow \dfrac{m}{x+m}=\dfrac{PB}{x}\Rightarrow PB=\dfrac{mx}{x+m}\\\\
AP+PB=x\Rightarrow\dfrac{nx}{x+n}+\dfrac{mx}{x+m}=x\\\\
\Longleftrightarrow\\\\
n(m+x)+m(x+n)=(x+n)(x+m)\\\\
\Longleftrightarrow\\\\
mn+nx+mx+mn=x^2+nx+mx+mn\\\\
\Longrightarrow
x^2=mn ~\Longrightarrow~ \bold{x= \sqrt{mn}}
[/tex]
