MathLabs

Problem 3

Each pair of opposite sides of convex hexagon has the property that the distance pp between their midpoints is 32\frac{\sqrt3}{2} times the sum of their lengths. Prove that the hexagon is equiangular.
Step 4 of 7: Force equality everywhere
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In plain words

The three non-positive gaps add up to a squared norm, so the norm must vanish.

2∣u−v+w∣2≤02|\mathbf u-\mathbf v+\mathbf w|^2\le0
Detailed analysis

Adding the three inequalities and moving the right side to the left gives 2∣u−v+w∣2≤02|\mathbf u-\mathbf v+\mathbf w|^2\le0. By Step 1 the squared norm is 00, so u−v+w=0\mathbf u-\mathbf v+\mathbf w=\mathbf0. Each of the three gaps used in Step 3 is non-positive and their sum is 00; consequently every one is 00, so all three triangle inequalities in Step 2 are sharp.