collinear points
Line up three friends so a fourth person sighting down the row sees only the nearest one, the others hidden behind. When points all sit on a single straight line like that, we call them collinear — literally 'on the same line'.
Precisely, a set of points is collinear if there exists one straight line that passes through every one of them. Any two points are always collinear, because two points already determine a line. The interesting case is three or more: points A, B, C are collinear exactly when C lies on line AB. If no single line holds them all, the points are non-collinear — and three non-collinear points are just what is needed to determine a plane and to form a triangle.
In coordinate geometry there is a quick test. Three points are collinear if the slope from the first to the second equals the slope from the second to the third — the path never bends. Equivalently, the 'triangle' they would form has zero area. Collinearity is the simplest order relation in geometry and underlies betweenness: only for collinear points does it make sense to ask whether one lies between the other two.
The points A(0, 0), B(2, 1), and C(4, 2) are collinear: the slope from A to B is 1/2, the slope from B to C is also 1/2, so all three lie on the same straight line y = x/2. By contrast A(0, 0), B(2, 1), D(2, 5) are non-collinear and form a genuine triangle.
Equal slopes between successive points signal collinearity; a triangle of zero area says the same.
Two points are always collinear, so the idea only bites for three or more; 'non-collinear' is the condition that makes a triangle and pins down a plane.