Suppose you have $6$ points $a_i\in\mathbb{R}^3$ $i\in\{1,..,6\}$ such that all triangles with vertices $0, a_i, a_{i+1}$ for $i\in\{1,..,5\}$ do not degenerate (I dont know if this assumption is really needed).
Trying different starting values, I found that the sum of the large cycle: $$ m = a_1 \times a_2 + a_2 \times a_3 + a_3 \times a_4 + a_4 \times a_5 + a_5 \times a_6 + a_6 \times a_1 $$ conincides with two smaller cylces: $$ t = a_1 \times a_3 + a_3 \times a_5 + a_5 \times a_1 + a_2 \times a_4 + a_4 \times a_6 + a_6 \times a_2 $$ where $\times $ is the usual cross product on the euclidean space $\mathbb{R}^3$. Can you help me to prove this conjecture? Or is there a simple counterexample (I have tried a bunch of examples and it always evaluated to $m=t$)?