Pappus' theorem
/ PAP-us /
Draw two straight lines and scatter three points anywhere on each. Now connect them up crosswise in a zig-zag to make a six-cornered path — a 'hexagon' whose corners alternate between the two lines. Pappus' theorem reveals a hidden order in this casual picture: the three points where opposite sides of the hexagon cross each other are themselves collinear. From two arbitrary lines and six freely chosen points springs one perfectly straight line, unbidden.
Precisely. Let A, B, C lie on one line and A', B', C' lie on another. Form the cross-connections and take three intersection points: P = (the meet of line AB' and line A'B), Q = (the meet of line AC' and line A'C), R = (the meet of line BC' and line B'C). Pappus' theorem says P, Q, R are collinear — they all lie on one line, called the Pappus line of the configuration. The result is purely about incidence, so it needs no measurement; in the projective plane it holds with no exceptions, parallel cases included (a 'missing' intersection simply sits on the line at infinity).
Pappus (Pappus of Alexandria, around 320 AD) predates the formal subject by over a thousand years, yet it is one of projective geometry's deepest facts. It is also self-dual — swapping points and lines returns the same theorem. Two honest remarks. First, Pappus is strictly stronger than Desargues: in the axiomatic theory, Pappus' theorem implies Desargues', but not conversely, and a projective plane satisfies Pappus exactly when its underlying coordinate system is a commutative field (so Pappus secretly encodes the commutativity of multiplication). Second, it is the degenerate, line-pair case of a far grander statement, Pascal's theorem on a hexagon inscribed in a conic — Pappus is what Pascal becomes when the conic splits into two straight lines.
On line 1 mark A, B, C; on line 2 mark A', B', C'. Draw AB' and A'B and mark their crossing P; draw AC' and A'C, crossing at Q; draw BC' and B'C, crossing at R. Lay a ruler on P and Q — it passes exactly through R as well. The three crossings are collinear, every time.
Six points on two lines, cross-joined: the three opposite-side intersections always line up.
Pappus is the conic-degenerated-to-two-lines case of Pascal's theorem, and it is logically stronger than Desargues' theorem (Pappus implies Desargues, not the reverse). Its holding is equivalent to the coordinate field being commutative — a surprisingly deep algebraic fact hiding inside a picture of dots and rulers.