The conditions dictate a specific hexagon the side of which can be determined.
Once the side is known the diagonal of the hexagon can be calculated.
The length from any 2 opposite vertices is 2 times the side and the incenter is the length of the side along this line.
The side of the hexagon and the angle which the point P makes with a vertex determine a pair of simultaneous equations based on the law of cosines. This pair can be solved for the angle and the side.
I will add those equations later when I have had an opportunity to check them again.
Draw a figure with a vertex of the hexagon at the origin and sides (length s) are y=x √3 and y=-x√3.
θ is the angle the line from P to the side y=-x√3.
The equations are:
256=64+s2-16s cos θ and
64=64+s2-16s cos (120-θ)
After some algebra, the solution is θ=90 and 120-θ=30.
Now s can be calculated from θ and from s the incenter.
