bonanova Posted February 25, 2015 Report Share Posted February 25, 2015 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. Quote Link to comment Share on other sites More sharing options...

0 gavinksong Posted February 25, 2015 Report Share Posted February 25, 2015 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. I had to think about this point: does it matter there are only n-1 rays? 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. Quote Link to comment Share on other sites More sharing options...

0 gavinksong Posted February 25, 2015 Report Share Posted February 25, 2015 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. Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted February 25, 2015 Author Report Share Posted February 25, 2015 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. I had to think about this point: does it matter there are only n-1 rays? Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted February 26, 2015 Author Report Share Posted February 26, 2015 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 n^{th} ray. That's a nice insight. 1 Quote Link to comment Share on other sites More sharing options...

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## bonanova

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

emanating from a vertex of a regular

-gon inscribed in the unit circle is 2nn.## Link to comment

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