Let
![A_0,A_1, \ldots , A_n](/media/m/9/8/a/98ad12b730feb030283a8dd8bb6cf766.png)
be points in a plane such that
(i)
![A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n](/media/m/9/6/7/967a99fee8c397f0e7f9a9237e232852.png)
and
(ii)
![0 < \measuredangle A_{0}A_{1}A_{2} < \measuredangle A_{1}A_{2}A_{3} < \cdots < \measuredangle A_{n-2}A_{n-1}A_{n} < 180^\circ](/media/m/5/2/8/528645a0e9be66f688dbfc092f6b633f.png)
,
where all these angles have the same orientation. Prove that the segments
![A_kA_{k+1},A_mA_{m+1}](/media/m/7/1/5/715c2b91c485a89a4dd48dfae01703a1.png)
do not intersect for each
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
such that
%V0
Let $A_0,A_1, \ldots , A_n$ be points in a plane such that
(i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and
(ii) $0 < \measuredangle A_{0}A_{1}A_{2} < \measuredangle A_{1}A_{2}A_{3} < \cdots < \measuredangle A_{n-2}A_{n-1}A_{n} < 180^\circ$,
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$