betweenness
Stand three people in a row along a hallway. We can all see at a glance who is in the middle — that everyday sense of one thing being 'between' two others is exactly what geometry needs to make precise so that proofs about order on a line can be trusted.
Betweenness is a relation among three collinear points: we say B is between A and C, written A-B-C, when all three lie on one line and B sits on the segment joining A and C. The clean numerical test, given the coordinate of each point along the line (from the ruler postulate), is that B is between A and C exactly when AB + BC = AC — the two short distances add up to the whole. If that equation fails, B is not between them. Note that A-B-C and C-B-A describe the very same situation; only the middle point's role is fixed.
Betweenness is the foundation of order on a line and the gateway to many later ideas: a segment AB is the two endpoints together with every point between them; B is the midpoint of AC when it is between A and C and AB = BC; a ray and the segment addition postulate both rest on it. Euclid used betweenness constantly from his diagrams but never stated axioms for it — a genuine gap that Hilbert later filled with explicit order axioms, though that repair belongs to a later field.
On a number line let A, B, C have coordinates 1, 4, 9. Then AB = 3, BC = 5, AC = 8, and since 3 + 5 = 8 we have AB + BC = AC, so B is between A and C. But for points at 1, 9, 4 in that naming, the middle by position is still the one at 4 — betweenness is about location, not the order you happen to list the names.
AB + BC = AC is the algebraic signature of B being between A and C.
Betweenness requires the three points to be collinear; for points off one line, 'between' has no meaning. Euclid assumed it from pictures without axioms, a gap Hilbert later closed.