Prove that if through three given points two planes can be drawn, then infinitely many planes throught these points can be drawn.

Question

Prove that if through three given points two planes can be drawn, then infinitely many planes through these points can be drawn. I don't get how this is possible, since there is unique plane passing through three point, how would I get two planes?

Answer 1

Let the three points be denoted $P,Q,R$. We wish to prove the following implication:

• If two planes exist containing $P,Q,R$, then infinitely many planes exist containing $P,Q,R$.

Let's break the proof into two cases.

Case 1: $P,Q,R$ are co-linear. Then the implication is true because its conclusion is true: infinitely many planes exist containing $P,Q,R$.

Case 2: $P,Q,R$ are not co-linear. Then the implication is true because its hypothesis is false: only one plane exists through $P,Q,R$.

Answer 2

Beware the hidden assumption."There is a unique plane passing through 3 points." This is only true if the 3 points are not co-linear. Don't assume the points are NOT co-linear. If 2 planes intersect, their intersection is a line, and there are infinitely planes containing that line. So if 3 points are co-linear, there are infinitely many planes through the line that contains the 3 points.