What counts as a theorem when distance is gone
By now you have rebuilt the plane on a new foundation. You added a horizon of points at infinity so that parallel lines finally meet, you learned to name every point by a triple of homogeneous coordinates, and in the last guide you met the principle of duality, where 'point' and 'line' trade places and every theorem acquires a twin. The projective plane you now live in has no distances, no angles, no notion of 'between'. So a fair question is: what is even left to prove? This guide answers with three of the most beautiful facts in all of geometry — two of them about pure incidence, one about a number that refuses to change.
The only relations projective geometry can speak about are incidence relations: which points lie on which lines, and which lines pass through which points. A statement like 'these three points are collinear' or 'these three lines meet in one point (concurrent)' is allowed; a statement like 'this segment is twice that one' is not, because doubling a length is meaningless once you can project the figure onto a slanted screen and stretch it. So a projective theorem is a guarantee that some incidences are forced by other incidences — true in every photograph of the figure, no matter how the camera is tilted.
Desargues: two triangles in perspective
Picture two triangles, ABC and A'B'C', sitting in the plane. Say they are in perspective from a point: there is a single point O — call it the center — such that line AA', line BB', and line CC' all pass through O. This is exactly what happens when you cast a shadow: a lamp at O throws triangle ABC onto a screen as A'B'C', and each vertex slides out along its own ray from the lamp. The vertices are tied together through one center.
Desargues' theorem says: whenever two triangles are in perspective from a point, they are also in perspective from a line. Concretely, pair up the sides. Side AB and side A'B' meet at some point P; side BC and side B'C' meet at a point Q; side CA and side C'A' meet at a point R. The theorem promises that P, Q, and R — three points that had no obvious reason to relate — are always collinear. They lie on one straight line, the so-called axis of perspectivity. Three concurrent lines through a center force three collinear meeting-points on an axis. That is the whole statement, and it is pure incidence: not a single length or angle appears.
Now collect the dividend from the previous guide. Desargues' theorem is almost perfectly self-dual. Read it forward: perspective-from-a-point implies perspective-from-a-line. Swap every 'point' for 'line' and 'lies on' for 'passes through', and you get the converse — perspective-from-a-line implies perspective-from-a-point — which is therefore true for free, with no new work. One theorem, proved once, hands you its own converse. That is duality paying real rent, not just decorating the wall.
Pappus: a hexagon woven from two lines
The second jewel is older — it goes back to Pappus of Alexandria around 320 CE, more than a millennium before projective geometry had a name, though we now see it as belonging here. Draw two distinct lines. On the first, mark any three points A, B, C; on the second, mark any three points A', B', C'. Now weave a zig-zag of six connecting segments between the two lines, crossing back and forth: A to B', B to C', C to A', and the three returning lines A' to B, B' to C, C' to A.
Pappus' theorem says these six lines cross in a hidden pattern. Pair them up by the crossings: line AB' meets line A'B at a point X; line BC' meets line B'C at a point Y; line CA' meets line C'A at a point Z. The claim — clean and surprising — is that X, Y, and Z are always collinear, lying on a single line no matter where you placed the original six points along their two lines. Six points split between two lines conjure a seventh line out of nothing. Like Desargues, it speaks only of meetings and collinearity; you could prove it with nothing but a straightedge.
It is worth pausing on how strict this is. Pappus' theorem is not merely a pretty pattern that usually works; it is forced, every time, with zero exceptions in the projective plane. And it carries a deep payload that you will only appreciate fully much later: Pappus' theorem holds in a coordinate plane precisely when the underlying number system has commutative multiplication. It is, astonishingly, the geometric face of the algebraic law a*b = b*a. We cannot prove that link at this rung, but it is honest to flag it: this innocent hexagon is secretly testing whether your arithmetic commutes.
The cross-ratio: the number projection can't kill
Now to the third jewel, and a confession built into projective geometry. Projection destroys almost every number you cared about. Take four points A, B, C, D in a row on a line and photograph them from a slant: lengths change, the ratio |AB| / |BC| changes, even the midpoint stops being the midpoint. It feels as though projection erases all numerical information. The astonishing discovery is that one number survives — and only one, in a precise sense. It is the cross-ratio, the single most important invariant in the whole subject. For four collinear points it is a ratio of ratios, built from signed lengths along the line, as the box below sets out.
AC / BC AC * BD
(A, B; C, D) = --------- = -----------
AD / BD AD * BC
(signed lengths: AC is positive one way, negative the other)
example on a number line, A=0 B=3 C=1 D=2 :
AC = 1, BC = -2, AD = 2, BD = -1
(A,B;C,D) = (1 * -1) / (2 * -2) = -1/(-4) = 1/4Here is the headline property, the reason the cross-ratio matters at all. If you project these four points from any center onto any other line, their cross-ratio is unchanged. Tilt the camera however you like; the four images carry the very same number 1/4. This is the fundamental invariant of projective geometry: cross-ratio is to projective geometry what distance is to Euclidean geometry — the quantity that all the allowed transformations agree to preserve. Four lines through a single point have a cross-ratio too (by duality, measured with the angles, or just by cutting them with any transversal and reading the four points), and it is the same whichever transversal you use.
When the cross-ratio equals exactly -1, the four points have a special name: D is the harmonic conjugate of C with respect to A and B, and the set is called a harmonic range. This is the projective shadow of 'C and D divide AB internally and externally in the same ratio', and it is everywhere — it is the configuration hiding inside a complete quadrangle, inside the relationship of pole and polar for a conic, and inside the midpoint together with the point at infinity. Harmonic conjugacy is the cross-ratio's most-used special case.
How the three jewels lock together
These are not three isolated curiosities; they brace each other. Desargues lets you compare two triangles through a center and an axis — the very setup of a perspectivity, which is one projection from a point. Chain two perspectivities together and you get a projectivity, the general map of projective geometry; and the cross-ratio is exactly the number every perspectivity, and hence every projectivity, preserves. Pappus, meanwhile, can be read as the statement that composing the right perspectivities behaves consistently. The three results are three angles on a single structure: the geometry of projection itself.
The cross-ratio's invariance also upgrades the whole subject from qualitative to quantitative. Desargues and Pappus tell you which incidences are forced; the cross-ratio gives you a real number you can actually compute and trust across any projection. That is the gateway to the fundamental theorem of projective geometry: a projective transformation of a line is pinned down completely by where it sends just three points, and once those three are fixed, the cross-ratio forces where every fourth point must go. Three points to set the stage, and cross-ratio to script the rest.
Let me be honest about depth, in the spirit of this ladder. We have stated all three theorems exactly and shown you faithful reasons to believe them — Desargues through the lift into space, the cross-ratio through a worked number — but airtight proofs from the projective axioms, and the deep link between Pappus and commutativity, belong to a dedicated course in projective geometry. What you should carry away is not a slogan but a working picture: in a world stripped of distance and angle, incidence still forces rich structure, and a single number rides through every projection unbroken.
One thread to carry into the final guide of this rung. The cross-ratio and harmonic conjugacy are not only about points on a line — they are the tools that tame curves. A conic in the projective plane can be defined entirely through cross-ratios of lines, with no mention of focus, eccentricity, or even the difference between an ellipse and a hyperbola. The last guide, Conics the Projective Way, walks straight through that door: the same machinery you built here turns the three classical conics into a single projective object.