$\alpha -\text{najmniejszy kąt ostry troˊjkąta prostokątnego}$  

$2\alpha -\text{pozostały kąt ostry troˊjkąta prostokątnego}$  

$\alpha +2\alpha +{90}^o={180}^o$  

$3\alpha ={90}^o$  

$\alpha ={30}^o$   

$2x+x\sqrt{3}=2\sqrt{6}+6\sqrt{2}$  

$x\left(2+\sqrt{3}\right)=\left(2\sqrt{6}+6\sqrt{2}\right)$  

$x=\frac{2\sqrt{6}+6\sqrt{2}}{2+\sqrt{3}}=\frac{\left(2\sqrt{6}+6\sqrt{2}\right)\left(2-\sqrt{3}\right)}{4-3}=$   

$=\left(4\sqrt{6}-2\sqrt{18}+12\sqrt{2}-6\sqrt{6}\right)=-6\sqrt{2}+12\sqrt{2}-2\sqrt{6}=6\sqrt{2}-2\sqrt{6}$  

$P=\frac{1}{2}\cdot x\cdot x\sqrt{3}=\frac{1}{2}\cdot \left(6\sqrt{2}-2\sqrt{6}\right)\left(6\sqrt{2}-2\sqrt{6}\right)\sqrt{3}=$  

$=\frac{1}{2}\cdot \left(72-12\sqrt{12}-12\sqrt{12}+24\right)\sqrt{3}=36\sqrt{3}-12\sqrt{36}+12\sqrt{3}=48\sqrt{3}-72=\underline{24\left(2\sqrt{3}-3\right)}$