incidence

Strip a plane of all measurement — no distances, no angles, no notion of which point is between which — and ask what bare facts survive. The answer is incidence: the simple relationship of a point 'lying on' a line, equivalently a line 'passing through' a point. It is the single primitive relation on which all of projective geometry is built, the only thing a shadow or a perspective photograph is guaranteed to preserve.

In projective geometry incidence obeys two crisp axioms that are perfectly symmetric: (1) any two distinct points are incident with exactly one common line — they determine a unique line through both; and (2) any two distinct lines are incident with exactly one common point — they meet in a unique point (this is the version of the parallel-free plane). In homogeneous coordinates incidence is a single equation: the point (x : y : z) is incident with the line [a : b : c] exactly when a x + b y + c z = 0. Notice the equation treats point-triple and line-triple alike, which is why it does not even matter, algebraically, which you call the point and which the line.

Because incidence is all projective geometry asks about, every projective theorem is ultimately a statement about which points lie on which lines — Desargues, Pappus, the construction of harmonic conjugates, the whole edifice. The symmetry of the two axioms is also the seed of the principle of duality. A caution: projective geometry deliberately keeps no other relation. 'This point is the midpoint', 'this angle is a right angle', 'this point is between those two' are not projective statements, because a projection can destroy them — only incidence is safe.