Question 159
Let $a_0, a_1, a_2, \dots$ be an arithmetic sequence of positive integers. A hexagon is called equiangular if all its internal angles have the same measure. Determine with proof for each $i \ge 0$ the number of incongruent equiangular hexagons the side lengths of which are the numbers $a_i, a_{i+1}, a_{i+2}, a_{i+3}, a_{i+4}, a_{i+5}$ in some order.
NOTE: A sequence $a_0, a_1, a_2, \dots$ is called an arithmetic sequence if $a_{i+1} - a_i = a_i - a_{i-1}$ for all $i \ge 1$.
NOTE: A sequence $a_0, a_1, a_2, \dots$ is called an arithmetic sequence if $a_{i+1} - a_i = a_i - a_{i-1}$ for all $i \ge 1$.