Let $v_0,v_1,\dots,v_d=v_0$ be a sequence of lattice points in $\mathbb Z^2$, in counterclockwise order (see figure below), such that successive pairs generate $\mathbb Z^2$ as a $\mathbb Z$-module.
I am trying to prove the following -
Show that if $d\ge4$ there must be two opposite vectors in the sequence, i.e., $v_j=-v_i$ for some $i,j$.
Suppose $v_i=-v_0$ and $i\ge3$, show that $v_j=v_{j-1}+v_{j+1}$
I have shown that $v_0,\cdots,v_d$ satisfy the following -
(a) $a_jv_j=v_{j-1}+v_{j+1}$ , $1\le i\le d$, for some integers $a_i$ and these integers satisfy, $$\left(\begin{array}{cc}0&-1\\1&a_1\end{array}\right)\left(\begin{array}{cc}0&-1\\1&a_2\end{array}\right)\cdots\left(\begin{array}{cc}0&-1\\1&a_d\end{array}\right)=\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$$
(b) Two cones $\langle v_i,v_{i+1}\rangle$ and $\langle v_j,v_{j+1}\rangle$ cannot be arranged with $v_j$ in the angle strictly between $v_{i+1}$ and $-v_i$ and $v_{j+1}$ in the angle strictly between $-v_i$ and $v_{i+1}$.
I know i have to use (b) somehow to prove 1 but I am unsure how. Help will be appreciated.
Thank you.