6. A point on a circle inscribed in a square is 1 and 2 units from the two closest sides of the square. Find the area of the square.
Solution: Call the point in question $A$, the center of the circle $O$, and its radius $r$. Consider a right triangle $B O A$ with hypotenuse $O A$ : $O A$ has length $r$, and $B O$ and $B A$ have lengths $r-1$ and $r-2$. By the Pythagorean theorem, $(r-1)^{2}+(r-2)^{2}=r^{2} \Rightarrow$ $r^{2}-6 r+5=0 \Rightarrow r=5$ since $r>4$. The area of the square is $(2 r)^{2}=100$.