Maximizing CD Vector Magnitude Minimizes Length of Path through points

First defining some terms and a hypothesis for closed paths P on the plane through n points with x,y coordinates x[i], y[i] all in the first quadrant:

c and d so named because they are the cross and dot products of vectors from the origin to successive points on the path, and l is the sum of the lengths of the lines between successive points on the path each squared...

For example I plotted c,d values for the 720 possible closed paths through 7 random points all with the same starting point and got:

P[1] and P[-1] were the two solutions to the travelling salesman problem, or paths minimizing l through the points with the starting point fixed, and they also were the two points that maximized c^2+d^2, P[-1] being just P[1] traversing the points in reverse order...

**It looked to me even more generally decreasing c^2 +d^2 increased l proportionally but I'll have to investigate further**