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

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

May 13, 2018
edited by Rollingblade  May 13, 2018

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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.

May 13, 2018
edited by GYanggg  May 13, 2018