Problem 3

Each pair of opposite sides of convex hexagon has the property that the distance

p

between their midpoints is

\frac{\sqrt3}{2}

times the sum of their lengths. Prove that the hexagon is equiangular.

Step 7 of 7: Conclude equal turns and equal angles

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In plain words

The equal long diagonals make the middle side direction the bisector of the first and third side directions.

\mathbf u=-(s+1)\mathbf b,\qquad \mathbf w=\left(1+\frac1s\right)\mathbf a,\qquad \angle(\mathbf a,\mathbf b)=\angle(\mathbf b,\mathbf c)=60^\circ

Detailed analysis

Using $\mathbf a+\mathbf c=s\mathbf b$ and $r=t=1/s$ , direct summation gives $\mathbf u=-(s+1)\mathbf b$ and $\mathbf w=(1+1/s)\mathbf a$ . Since $|\mathbf u|=|\mathbf w|$ and $\angle(\mathbf u,\mathbf w)=120^\circ$ , the angle between $\mathbf a$ and $\mathbf b$ is $60^\circ$ . Also $|\mathbf a|=s|\mathbf b|=|\mathbf a+\mathbf c|$ ; hence $|\mathbf c|=|\mathbf a|$ from $\mathbf c=s\mathbf b-\mathbf a$ , and $\mathbf a+\mathbf c=s\mathbf b$ makes the angle between $\mathbf b$ and $\mathbf c$ also $60^\circ$ . The opposite sides are antiparallel, so the six successive turns are all $60^\circ$ and every interior angle is $120^\circ$ .