2022 AMC 12B Problem 24

A chord joining two vertices $d$ steps apart has squared length $2 - 2\cos\dfrac{2\pi d}{7},$ and there are $7$ chords for each of $d = 1, 2, 3.$ The required sum is $$7 \sum_{d=1}^{3} \left(2 - 2\cos\tfrac{2\pi d}{7}\right)^2.$$ Using $\displaystyle\sum_{d=1}^{3} \cos\tfrac{2\pi d}{7} = -\tfrac12$ and $\displaystyle\sum_{d=1}^{3} \cos^2\tfrac{2\pi d}{7} = \tfrac54,$ the inner sum expands to $4\left(3 + 1 + \tfrac54\right) = 21.$ Therefore the total is $7 \cdot 21 = 147.$ Thus, the correct answer is C.