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 6 of 7: Use sharpness to orient opposite sides
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In plain words

Equality in the triangle inequality makes each opposite pair parallel in opposite traversal directions.

a+c=sb,DE→=−ra,EF→=−sb,FA→=−tc,r=t=1s\mathbf a+\mathbf c=s\mathbf b,\qquad \overrightarrow{DE}=-r\mathbf a,\quad\overrightarrow{EF}=-s\mathbf b,\quad\overrightarrow{FA}=-t\mathbf c,\qquad r=t=\frac1s
Detailed analysis

Put a=AB→\mathbf a=\overrightarrow{AB}, b=BC→\mathbf b=\overrightarrow{BC}, and c=CD→\mathbf c=\overrightarrow{CD}. Sharpness in Step 2 gives positive r,s,tr,s,t with DE→=−ra\overrightarrow{DE}=-r\mathbf a, EF→=−sb\overrightarrow{EF}=-s\mathbf b, and FA→=−tc\overrightarrow{FA}=-t\mathbf c. The zero-sum relation in Step 1 is equivalent to a+c=sb\mathbf a+\mathbf c=s\mathbf b. The side-vector closure, together with this relation and the non-parallel vectors a,c\mathbf a,\mathbf c, gives r=t=1/sr=t=1/s.