therightwomanwf

2022-07-11

Given two noncoplanar lines p and q, and a point A, does there always exist a line that passes through p, q and A?

treccinair

Expert

Step 1
Once you have the plane $\alpha \left(p,A\right)$ , this happens if q belongs to a plane which is parallel to $\alpha$ .
For example, let p be the x axis and $A=\left(0,1,0\right)$ : the plane containing them is $z=0$ . Any line passing through p and A is confined on this plane (axiom I.6: "the line lies on the plane"). Now imagine q being on the plane $z=1$ and you have your counterexample.

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