Lines and planes - general concepts

Question

I've come across a book that has this general questions about lines and planes. I can't agree with some of the answers it presents, for the reasons that I'll state below:

True or False:

• Three distinct points form a plane - BOOK ANSWER: True - MY ANSWER: False, they cannot belong to the same line

• Two intersecting lines form a plane - BOOK ANSWER: True - MY ANSWER: False, they can be parallel and coincident lines.

• Two lines that don't belong to a same plane are skew - BOOK ANSWER: True - MY ANSWER: True

• If three lines are parallel, there is a plane that contains them - BOOK ANSWER: True - MY ANSWER: False, they can be parallel and coincident lines.

• If three distinct lines are intersecting two by two, then they form only one plane - For this last one there's no answer and I'm not sure about the conclusion.

If you could help me, I appreciate it.

Thank you.

Answer 1

Assuming 3D Euclidean geometry:

• On point 1 if the three points are collinear there is more than one plane

• I would agree with the book and disagree with you on point 2 for a suitable definition of intersecting (at exactly one point).

• I would disagree with the book on point 4 (consider the three parallel edges of a triangular prism).

• I would say True for point 5, again for a suitable definition of intersecting two-by-two (the plane is defined by the three points of intersection, which do not lie on a single line and all three lines lie on it)

Answer 2

(1) For three points to be on a plane, they need not be on the same line. Draw a triangle on a sheet of paper, and you will see what I mean.

(2) If they are coincident lines, they are not called 'intersecting' lines. Intersecting lines meet at one point ONLY.

(4) You are right. (However, even if they were all coincident they would still belong to some plane). But, in general they need not belong to a plane.

(5) True. Referring to (2), intersecting lines lie in a plane. If line A intersects line B, then A and B are in same plane. If B intersects with line C, then B and C are in same plane. Thus, intersection of A and C should also be in the same plane.