A 14. Triangles in a regular hexagon

I came across a neat problem in Quora. When I found it, there were no solutions, and with crucial help from my student Apratim Roy, I was able to find a very elegant solution. Unfortunately, when I went back to Quora I was unable to find the original problem, so I cannot give credit to the proposer. If anyone wants, send me an email and I will send you my solution, or provide a hint.

Here is the problem: Let ABCDEF be a regular hexagon, and P a point inside the hexagon. Suppose Area(PAB) = 3, Area(PCD) = 5, and Area(PEF) = 8. Find Area(PBC).