From the diagram
[tex]e+d=c\text{ ---- equation 1}[/tex][tex]\sin \alpha=\frac{a}{c}[/tex][tex]\cos \beta=\frac{d}{a}[/tex]
[tex]\sin \text{ }\beta=\frac{b}{c}[/tex][tex]\cos \alpha=\frac{e}{b}[/tex]
[tex]\begin{gathered} \sin \alpha=\sin (90-\beta) \\ \sin \alpha=cos\beta\text{ } \\ \frac{a}{c}=\frac{d}{a} \end{gathered}[/tex][tex]a^2=cd----\text{ equation 2}[/tex]
[tex]c^2=a^2+b^2[/tex][tex]\begin{gathered} \sin \beta=\cos \alpha \\ \frac{b}{c}=\frac{e}{b} \end{gathered}[/tex][tex]b^2=ce-----\text{ equation 3}[/tex]
[tex]\begin{gathered} \text{from equation 2 and 3} \\ a^2+b^2=cd+ce \end{gathered}[/tex][tex]a^2+b^2=c(d+e)[/tex]
If equation 1 is factored out
then
[tex]a^2+b^2=c(c)[/tex][tex]a^2+b^2=c^2[/tex]
