MathLabs

Problem 3

In an nn-gon all interior angles are equal, and the lengths of consecutive sides satisfy a1≥a2≥⋯≥ana_1\ge a_2\ge\cdots\ge a_n. Prove that a1=a2=⋯=ana_1=a_2=\cdots=a_n.
Step 1 of 5: Record the common turning angle
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In plain words

The polygon has a rigid angular pattern; only the side lengths could break its symmetry.

θ=2πn\theta=\frac{2\pi}{n}
Detailed analysis

A convex equiangular polygon turns through the same exterior angle θ=2π/n\theta=2\pi/n at every vertex. Thus the directions of successive sides are fixed rotations of one another.