Problem 3

In an

n

-gon all interior angles are equal, and the lengths of consecutive sides satisfy

a_1\ge a_2\ge\cdots\ge a_n

. Prove that

a_1=a_2=\cdots=a_n

Step 5 of 5: Recover the first side from closure

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In plain words

Once all remaining sides have the same length, their direction vectors cancel to the opposite of the first unit direction.

a_1e_0+t(e_1+\cdots+e_{n-1})=(a_1-t)e_0=0\Longrightarrow a_1=t

Detailed analysis

Let $e_0,e_1,\ldots,e_{n-1}$ be the unit vectors in the successive side directions. Since $e_0+\cdots+e_{n-1}=0$ and $a_2=\cdots=a_n=t$ , closure becomes $a_1e_0+t(e_1+\cdots+e_{n-1})=(a_1-t)e_0=0$ . Hence $a_1=t$ , so every side length is equal.