a plane
Imagine the surface of a perfectly still pond, or an infinitely large, perfectly flat tabletop with no edges. A plane is that idea: a flat surface that has length and width but no thickness, stretching out forever in every direction within itself.
A plane is the third undefined term of Euclidean geometry, alongside point and line, and again the axioms tell us how it behaves rather than what it is. Three points that do not all lie on one line determine exactly one plane — this is why a three-legged stool never wobbles, while a four-legged one can. A plane is two-dimensional: a point in it is fixed by two numbers, as the coordinate pair (x, y) shows. If two points lie in a plane, the entire line through them lies in that plane as well, and two distinct planes either never meet or meet along a single straight line.
Like its cousins, a plane drawn on paper is shown as a tilted parallelogram with a name such as plane P or plane ABC, but that shape is only a window onto an unbounded flat sheet. Points that all lie in one plane are called coplanar; figures studied entirely within a single plane belong to plane geometry, while figures using more than one plane belong to solid geometry.
A camera tripod stands steady because its three feet, as three non-collinear points, determine exactly one plane to rest on. A fourth foot would need to land in that same plane by luck, which is why four-legged tables so often rock.
Three non-collinear points pin down one plane; that is why three legs never wobble.
Three points fix a plane only if they are not collinear; three points on one line lie in infinitely many planes, like the pages of a book sharing one spine.