>>12296624To complete this autism, let us characterize a-f themselves. Clearly each of the two sides a+b+c or d+e+f are manifestly between 0 and 200. The lengths a, b, and c are clearly in a 1/2 : 1/6 : 1/3 relationship, adding to a unit. The d + e + f side is a bit subtler, being in a 1/4 : 9/20 : 3/10 relationship, again adding to a unit. The choice of any one length (within its strict postive range, each of which can be worked out and are left as an exercise) fixes the other five.
Consider the "natural" choice of lengths for a, b and c (picking any one of them instantly fixes the other two, and the other side d-f in turn): a = 3, b = 1, c = 2. This immediately entails d = 25/3, e = 15, f = 10 (see middle row in diagram). In the other case, the "natural" choice is the distinct d = 5, e = 9, f = 6 which fixes a = 5, b = 5/3, c = 10/3, where the appropriate sums of triplets and products in each case always give 200 as a check.
It may be an abuse of notation to describe the two sides as "linearly independent", but the natural language of the phrase is appropriate here. I herewith deliver myself of this middle school-tier recreation.
