the principle of duality
Read any theorem of plane projective geometry and you will find it talks about points and lines, joined only by incidence. The principle of duality is the startling observation that if you take such a true theorem and swap every 'point' with 'line', every 'lies on' with 'passes through', every 'the line joining two points' with 'the point where two lines meet', the new sentence is also a true theorem — its dual — for free, with no separate proof. Buy one theorem, get a second one at no cost.
Why does this work? Because the entire axiom system of the projective plane is symmetric in points and lines. The two governing axioms — 'two points determine one line' and 'two lines determine one point' — turn into each other under the swap, and so does the incidence equation a x + b y + c z = 0, which treats the point-triple (x : y : z) and the line-triple [a : b : c] identically. So any chain of reasoning built only from these can be transcribed term-by-term into a valid dual chain. The dictionary to apply is short: point becomes line, line becomes point, collinear (points on a common line) becomes concurrent (lines through a common point), 'the join of two points' becomes 'the intersection of two lines', and a configuration of p points and q lines becomes one of q points and p lines.
Duality is not a vague analogy; it is an exact, mechanical correspondence, and it roughly halves the labour of the subject. The dual of Desargues' theorem is its own converse; the dual of 'three points are collinear' is 'three lines are concurrent'. Two honest cautions. First, duality lives in the projective plane: it relies on the parallel-free incidence axioms, so it does not hold cleanly in ordinary Euclidean geometry. Second, the dual of a true statement is true, but the dual of a particular figure you drew is a different figure — duality swaps the roles, it does not say a point 'is' a line.
Statement: 'Two distinct points lie on exactly one common line.' Apply the dictionary — swap point and line — and you get: 'Two distinct lines pass through exactly one common point.' Both are axioms of the projective plane, and each is the dual of the other. Likewise the dual of 'three points are collinear' is 'three lines are concurrent'.
Swap point and line, collinear and concurrent — a true theorem maps to a true dual theorem.
Duality is special to projective geometry; it fails in affine or Euclidean geometry because there the parallel exception breaks the symmetry of the incidence axioms. Also, a self-dual theorem (one identical to its own dual, like Desargues including its converse) is not a contradiction — it just means the swap returns the same statement.