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June 11th, 2011, 07:19 AM  #1 
Newbie Joined: Jun 2011 Posts: 1 Thanks: 0  regular polygon
choose some point inside a regular polygon and connect it to every vertex. the sum of length of these line segments achieves minimum value when the point chosen is the center of the polygon. can someone give a proof of it? thanks a lot~~~ 
June 16th, 2011, 07:07 AM  #2 
Senior Member Joined: Jun 2011 Posts: 298 Thanks: 0  Re: regular polygon
We can prove this by mathematical induction on integer , where is the number of vertices on the polygon. The smallest is 3, i.e. a triangle, and the largest is an gon. To prove by induction, we must prove that it's true for a triangle; then prove that it's true for gon. To prove a triangle, you must know that the center is 1/3 of the height from the base. I will leave it to you. Since polygon of 4 sides is easy to prove, I will show here: Since a regular polygon has equal sides, a 4 polygon is a square. The center is where the two diagonal lines intersect. To make it simple, let the length of each side be . There are two extreme cases where you can pick a point. The 1st one is the center point, and the other is located right on one of the vertices. 1st extreme, center point, by Pythagoras theorem, we know that the length of each diagonal is , so the sum of these two segments is . The 2nd extreme, on a vertex. The sum of the segments from this point is 2 sides of the square plus a diagonal. In other words, . In all cases, the sum will lie between the 1st extreme and the 2nd extreme. 

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