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# Squared rays

## Question

Show that the sum of the squares of the lengths of all sides and diagonals

emanating from a vertex of a regular n-gon inscribed in the unit circle is 2n.

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The square lengths of the xth ray can be expressed using the law of cosines: 2 - 2 cos(x * 2pi / n).

If we sum this over all the rays, the cosine terms cancel out, so we are left with 2n.

That's an excellent question.

We include an imaginary 0th ray, the square length of which is 2 - 2 cos(0) = 0.

This is necessary to have all the cosine terms cancel each other out.

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The square lengths of the xth ray can be expressed using the law of cosines: 2 - 2 cos(x * 2pi / n).

If we sum this over all the rays, the cosine terms cancel out, so we are left with 2n.

##### Share on other sites

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The square lengths of the xth ray can be expressed using the law of cosines: 2 - 2 cos(x * 2pi / n).

If we sum this over all the rays, the cosine terms cancel out, so we are left with 2n.

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Yeah, I had to think that point through for a moment.

I saw it as having a sum that I didn't know how to evaluate.

So I added and subtracted the term that would make the sum zero.

That step simultaneously evaluated the sum and added 2 to the mix.

Your approach is neater, reasoning that the answer is not changed if you add the squared length of the nth ray.

That's a nice insight.

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