Thanks for the ideas.  Your leisure time is well spent!
It seems to me that the fact that three of the five points are identical
means that my CAD program ought to use a triangle primitive instead of a
polygon.  That way there will be no need to test for coplanarity, which it
seems to me is _bound_ to fail due to rounding in some circumstances
(except in the trivial cases of the points all lying on a single x, y or x
plane).
So I'll try to speak soothingly to the writer of the CAD program...

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