Given a polygon with known perpendicular lines from any point inside the polygon, make a statement about the side length.

Hi Rollingblade!

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. 

\(S_{abcdef}=\frac12m(a+b+c+d+e+f)\)

The area can also be expressed as the sum of the areas of the 6 equilateral triangles. 

\(S_{abcdef}=\frac12\cdot6\cdot{m}\cdot{h}\)

Since these two expressions are equal, we get:

\(6h=a+b+c+d+e+f\)

This is true for any polygon. 

n-sided polygon:

\(n\cdot{h}=\overbrace{a+b+c+d+\dots+x+y+z}^\text{n-sides}\)

Where h is the height, and a - z are the known perpendicular lines from any point.