Wolfram Language (Mathematica), 32 bytes
This is a port of Noodle9's Python answer. I start by determining the centroid of the four points, which is their average position in space, by taking the mean of the four vectors corresponding to each. [N.B.: If the points had mass, and their masses were identical, this would ...
Vyxal, 12 bytes
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ƛ ; # Map to...
? # Input
ÞT # Transposed
vṁ # Take the average of each (center)
- # Get the x and y distances to the center
²∑√ # Pythagoras to get abs. dist. to center
≈ # Are all equal?