Choose some point A inside a regular polygon. The sum of length of line segments connecting A to the vertices of the polygon achieves minimum when A is the center of the polygon.
Can somebody give a proof of this? Thanks!!
Is it valid? Your personal exploration would be of most benefit to you.
Prove it for a triangle. Does it work?
Pove it for a square. A Rhumbus. A convex quadrilateral. A concave quadrilateral. (That last one presents some LOS problems.)
What say you? Do you believe it should be provable?
EDIT: late in 2 minutes.
Is it enough to prove the following for making the conclusion in OP's question?
Say we have regular polygon A1A2...An with center O. Let denote for . Then
and here's the proof using vectors for Zarathustra's question, in the attachment. Interestingly, I suddenly realize that this together with the the argument in #5 also allows us to compare the sums of connections for any two points in the polygon: the one farther away from the center O has a larger sum. And the same comparison can be made even when one or both points are outside the polygon! (since there's no requirement in #5 and #12 for A to be inside the polygon)
The problem has a nice and simple structure.
EDIT: the contours are not circles... so we cannot compare arbitary points. The best #5 and #12 allow us is to compare 2 points if they and the center O lie on a straight line.