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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.

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