Constant Distance of Trig Points

Geometry Level 4

The distance between the points \[\left(\dfrac{\sqrt{3}}{2}\sin (\alpha-c), \dfrac{1}{2}\sin(\alpha-c)\right)\] \[\left(\dfrac{1}{2}\sin (\alpha+c), \dfrac{\sqrt{3}}{2}\sin(\alpha+c)\right)\]

is constant, for some constant \(c\) and variable \(\alpha\).

If the distance between the points can be expressed as \(\dfrac{p}{q}\) for positive coprime integers \(p,q\), then find \(p+q\).

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