Prove the sum the projections of the sides of a polygon on any line is zero

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Prove the sum the projections of the sides of a polygon on any line is zero.

I've been trying to learn trigonometry on my own and to develop my math skills when I can across this problem.

I can visualize how the polygon can be constructed from vectors that follow along its perimeter.

These vectors, because they are changing direction and are projected onto a line that has a single direction, would result in the sums of the projections on a line to be zero (I think).

How do I come up with a convincing proof for this problem.