constructing an equilateral triangle in the Beltrami klein model

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I am puzzeling with the following:

Using the beltrami klein disk of hyperbolic geometry (see https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model ) (PS not the poincare disk model)

and given a segment inside it

How to construct an equilateral triangle where the segment is one of its sides?

I know you could first translate the endpoints to a Poincare disk points , do the construction the poincare disk and then translate the third point back to the beltrami model.

But is there no simpler way?