A question for university students and professors about polygons and their properties.?

Question

Suppose that we have a regular polygon with 'n' sides. 'M' is an arbitrary point in this polygon whose distance from the sides of the polygon are 'x_1', 'x_2', ..., 'x_n'.

Show that this equation holds:
http://m1.freeshare.us/167fs644292.gif

where 'a' is the length of each side.

Answer 1

Answer:
Σ 1/x_i > Σα_i/a = 2π/a,
where α_i is anglular size of side "i" as seen from point M.



Consider arbitrary triangle ABC, angle α = BAC. Lets denote by X the closest point to vertex A on side BC. Distance from A to side BC is x = |AX|. Area S of the triangle is

S = 1/2 ax sin(angle between AX and BC) <= 1/2 ax

Draw a circle centered at A and radius x.
Intersection of this cicrle and triangle is sector of area 1/2 αx², and is less than area S of the whole trangle:

1/2 αx² < S <= 1/2 ax,

thus for arbitrary triangle we have 1/x > α/a

Answer 2

Since the polygon is regular, we can inscribe a circle inside it; let the radius of this circle be r. The distance from any point inside the polygon to any side will be less than the diameter of the circle = 2r; that is to say, x_n < 2r for any n. Invert these to have (1/x_n) > 1/2r, and now sum all the distances.

Sum = S > n*(1/2r) = (n/2r).

The circumference of the circle = 2*pi*r < n*a (the number sides * the length of each side) --> (n/2r) > (pi/a).

This gives us: S > (pi/a).

*sigh* That is as close as I've gotten thus far; I am figuring out where the other factor of 2 arises to give S > (2*pi)/a.

Answer 3

Where on the sides?
The middle?
Or do you mean vertices?