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 5 of 7: Extract equal diagonal lengths
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In plain words

The equality equations and the zero-sum relation leave an equilateral three-vector configuration.

∣u∣=∣v∣=∣w∣,∠(u,w)=120∘|\mathbf u|=|\mathbf v|=|\mathbf w|,\qquad \angle(\mathbf u,\mathbf w)=120^\circ
Detailed analysis

From u−v+w=0\mathbf u-\mathbf v+\mathbf w=\mathbf0 we have v=u+w\mathbf v=\mathbf u+\mathbf w. The third equality from Step 3 is ∣u∣2+∣w∣2=−4u⋅w|\mathbf u|^2+|\mathbf w|^2=-4\mathbf u\cdot\mathbf w, so ∣v∣2=(∣u∣2+∣w∣2)/2|\mathbf v|^2=(|\mathbf u|^2+|\mathbf w|^2)/2. Substitution in the first equality simplifies to 3(∣w∣2−∣u∣2)=03(|\mathbf w|^2-|\mathbf u|^2)=0. Thus ∣u∣=∣w∣|\mathbf u|=|\mathbf w|, then ∣v∣=∣u∣|\mathbf v|=|\mathbf u|; moreover u⋅w=−∣u∣2/2\mathbf u\cdot\mathbf w=-|\mathbf u|^2/2, so ∠(u,w)=120∘\angle(\mathbf u,\mathbf w)=120^\circ.