Question 93
Let $A_{0}, A_{1}, \ldots, A_{n}$ be points in a plane such that
(i) $A_{0}A_{1} \leq \frac{1}{2}A_{1}A_{2} \leq \cdots \leq \frac{1}{2^{n-1}}A_{n-1}A_{n}$ and
(ii) $0 \lt \measuredangle A_{0}A_{1}A_{2} \lt \measuredangle A_{1}A_{2}A_{3} \lt \cdots \lt \measuredangle A_{n-2}A_{n-1}A_{n} \lt 180^{\circ}$,
where all these angles have the same orientation. Prove that the segments $A_{k}A_{k+1}, A_{m}A_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m-2 \lt n-2$.
(i) $A_{0}A_{1} \leq \frac{1}{2}A_{1}A_{2} \leq \cdots \leq \frac{1}{2^{n-1}}A_{n-1}A_{n}$ and
(ii) $0 \lt \measuredangle A_{0}A_{1}A_{2} \lt \measuredangle A_{1}A_{2}A_{3} \lt \cdots \lt \measuredangle A_{n-2}A_{n-1}A_{n} \lt 180^{\circ}$,
where all these angles have the same orientation. Prove that the segments $A_{k}A_{k+1}, A_{m}A_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m-2 \lt n-2$.