Points equally spaced along a circle, connected by lines to a common point

Consider the unit circle. Draw equally spaced points along its circumference. Draw lines from the rightmost point to each of the other points. What’s the sum and product of the lengths of the lines?

Example: n = 3 points

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Equation of unit circle: x² + y² = 1

Coordinates in terms of θ. Polar coordinates: x = cos θ, y = sin θ

θ angle separation between points is the same: 2π/n.

Coordinate of rightmost point: x=1, y=0;

Coordinate of neighbor at θ = 2πm/n: x = cos 2πm/n, y = sin 2πm/n

Length of line between the two points:

lₘₙ = √{(x₁ - x₂)² + (y₁ - y₂)²} 
... = √{(cos(2πm/n) - 1)² + sin(2πm/n)²}
... = √{cos(2πm/n)² - 2 cos(2πm/n) + 1 + sin(2πm/n)²}
... = √{2 - 2 cos(2πm/n)}
... = √{4 sin^2((m π)/n)} 
... = 2 sin(mπ/n)

Product of the lengths:

Π m ∈ 1..n-1, 2 sin(mπ/n)

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Wolframalpha couldn’t simplify. Can we find another expression that’s not in terms of the individual lengths?

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e^{iθ} = cos θ + i sin θ.

x-y coordinates.

Length of line between rightmost point and point at θ: cos θ - 1 + i sin θ = e^{iθ} - 1

The product of the lengths of the lines for three points:

(1 - e^{i2π/3})(1 - e^{i4π/3}) = 1 - e^{i4π/3} - e^{i2π/3} + e^{i6π/3} 
                           ... = 2 - e^{i2π/3} - e^{i4π/3}
                           ... = 2 - ∑ m ∈ 1..2, e^{i2mπ/3}

So if we do it this way we get a sum instead of a product.

The product of the lengths of the lines for four points:

(1 - e^{i2π/4})(1 - e^{i4π/4})(1 - e^{i6π/4}) = 1 - ...

Try a rewrite: Let z := e^{i2π/4}.

Product becomes

(1-z)(1-z²)(1-z³) = ...

∑ n ∈ 1..m, zⁿ?

z * (z^m - 1) / (z - 1)

Π m ∈ 1..n, zⁿ?

log then de-log

log (Π n ∈ 1..m, zⁿ) = (log z)(∑ n ∈ 1..m, n) = log(z)(m(m+1)/2) = log(z^(m(m+1)/2)).

So Π n ∈ 1..m, zⁿ = z^(m(m+1)/2).

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(x-z)(x-z²)(x-z³).. is a polynomial. We evaluate it at x = 1.

Let’s write out the first few terms. Can we find a pattern?

(x² - z²x - zx + z³)(x - z³) = (x³ - x²z³ - z²x²+ x z^5 -zx² + z^4x + xz³ - z^6)

For n points, z^n = 1.

Simplify:

(x^3 - x^2z^3 - z^2x^2+ x z -zx^2 + x + xz^3 - z^2)