Four Points in the Plane

Find all configurations of four points in the plane, such that the pairwise distances between the points take at most two different values.

All 5 configurations are shown below: a square, a rhombus with 60°-120°-60°-120°, an equilateral triangle with its center, an isosceles triangle with 75°-75°-30° and its center, and a quadrilateral with 75°-150°-75°-150°.

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