As we did with Euclidean motions, we can use Poincaré motions to simplify proofs or calculations by moving the given configuration of points, lines, etc. to a more convenient location. (Note the form of the motion formulas:

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It's easy to move any particular point to the origin.) Here's a numerical example you did directly earlier. Now try to do it by first moving the points so that one of them moves to the origin.