Hexacircle

In a hexagon that is tangential to a circle and that has consecutive sides $a, b, c, d, e,$ and $f$ , we have the important relation $a+c+d = b+e+f$ . Hence, $2+5+9 = 3+7+x \Rightarrow x = \boxed{6}$ . I didn't feel it worth mentioning the proof of this, as it has already been demonstrated in Sir Niranjan Khanderia's solution and is quite a trivial deduction from tangent properties.

Two tangents meeting at a point, have equal lengths. The length of a side is the sum of the length of tangent to the left and to the right from the point. of contact with the circle. The length to the right of a point is therefore equal to the length to the left of the next point( clockwise order. ) So if the first side is 2, $\color{#D61F06}{\text{first side will have 1 on its left }}$ and have and1 on its right. . So the next side 3 will have 1 on its left, and so 2 to its right. Side 5 will have 2 to its left, so 3 to its right, And so on. The 9 side is the fifth. $\color{#3D99F6}{Fifth~ side~ will~ have}$ 4 to its left and $\color{#3D99F6}{5 ~to~ its~ right}$ . The sixth side will therefore have 5 to the left. Next point is first point having length 1 to it left. So the sixth side will have 1 to itsright.. So sixth side is 5 + 1 = 6 long.