Melody made a similar answer here: https://web2.0calc.com/questions/weekly-cycle-challenge-2-help
I was thinking about Melody's answer and was thinking about how to generalize it. Here is what I have:
Using the hexagon problem:
The area of the hexagon is equal to the sum of the six triangles constructed from connecting the point inside the hexagon to the 6 vertices.
The area can also be expressed as the sum of the areas of the 6 equilateral triangles.
Since these two expressions are equal, we get:
This is true for any polygon.
Where h is the height, and a - z are the known perpendicular lines from any point.