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 1 of 7: Set the three long-diagonal vectors
Previous stepNext step
In plain words

Pair the three opposite-side conditions into three long-diagonal vectors whose signed sum telescopes to zero.

u=DA→,v=EB→,w=FC→,u−v+w=0\mathbf u=\overrightarrow{DA},\quad\mathbf v=\overrightarrow{EB},\quad\mathbf w=\overrightarrow{FC},\qquad \mathbf u-\mathbf v+\mathbf w=\mathbf0
Detailed analysis

For ABCDEFABCDEF, define u=DA→\mathbf u=\overrightarrow{DA}, v=EB→\mathbf v=\overrightarrow{EB}, and w=FC→\mathbf w=\overrightarrow{FC}. Expanding the six side vectors gives u−v+w=0\mathbf u-\mathbf v+\mathbf w=\mathbf0; this is just telescoping, not an extra geometric assumption.