the fundamental theorem of projective geometry
How much freedom does a projective transformation have, and how much do you need to know to specify one completely? The fundamental theorem of projective geometry gives the clean answer for the line and for the plane, and it is the result that makes projective transformations genuinely usable: it says they are rigid in exactly the right amount — pinned down by a small, sharp number of points.
On a projective line the statement is: there is exactly one projectivity sending any three distinct points to any three distinct points. Three in, three out, and the map is forced everywhere else — in particular it must preserve the cross-ratio of any fourth point with the three. In the projective plane the matching statement is: given any four points in general position (no three collinear) and any other four points in general position, there is exactly one collineation carrying the first four to the second four, in order. So a plane collineation has the freedom of four points and not one degree more. The proof reduces to linear algebra in homogeneous coordinates: choosing where four suitably independent points go fixes the 3-by-3 matrix up to the harmless overall scaling.
This theorem is the backbone of the whole subject. It tells you cross-ratio is a complete invariant on a line (two configurations of four points are projectively the same exactly when their cross-ratios agree), it underwrites Steiner's projective definition of a conic, and it guarantees the constructions of projective geometry are well-defined. One honest qualification, the same one that haunts collineations: the theorem as stated — every cross-ratio-preserving map is a matrix projectivity — is true over the real numbers, but over a field with extra automorphisms (the complex numbers, for example) one must allow 'semilinear' maps too. So the sharpest form of the theorem is stated for the real (or rational) projective space; this is not a defect but a precise statement of where the clean picture lives.
On a line, suppose a projectivity must send 0 -> 1, 1 -> 2, and infinity -> 3. The theorem says these three assignments determine it uniquely; the map turns out to be x -> (something) and, whatever it is, it must send any fourth point so as to preserve cross-ratio. In the plane, naming the images of four points like (1:0:0), (0:1:0), (0:0:1), (1:1:1) fixes the entire 3-by-3 matrix.
Three points fix a projectivity of the line; four points in general position fix a collineation of the plane.
The crisp form 'every line-preserving bijection is a matrix projectivity' holds over the reals; over fields with nontrivial automorphisms (e.g. the complex numbers) one must also allow semilinear maps. Also, 'general position' is essential: four points with three collinear do NOT determine a unique collineation, so the genericity hypothesis cannot be dropped.