To solve the problem of finding the sum of squares of the lengths of all sides and diagonals of a regular hexagon inscribed in a unit circle, follow these steps:
Step 1: Identify the types of segments
A regular hexagon has 6 vertices on the unit circle. The segments between vertices are:
- Adjacent sides: Distance between vertices with a gap of 1 (e.g., 0-1).
- Diagonals with gap of 2: Distance between vertices with a gap of 2 (e.g., 0-2).
- Diagonals with gap of 3: Diameters (e.g., 0-3).
Step 2: Calculate square of distances
For the unit circle:
- Adjacent sides: Square of distance = $1^2 = 1$ (since distance = $2\sin(\pi/6)=1$).
- Gap of 2: Square of distance = $(\sqrt{3})^2 = 3$ (distance = $2\sin(\pi/3)=\sqrt{3}$).
- Gap of 3: Square of distance = $2^2 = 4$ (distance = 2, diameter).
Step 3: Count each segment type
- Adjacent sides: 6 segments.
- Gap of 2: 6 segments (each vertex contributes one).
- Gap of 3: 3 segments (diameters).
Step 4: Sum the squares
Total sum = $(6 \times 1) + (6 \times 3) + (3 \times 4) = 6 + 18 + 12 = 36$.
Answer: $\boxed{36}$
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