In the projective plane a point (a,b,c) and a line ax+by+cz=0 are the same triple; incidence — point lies on line — is the symmetric condition a₁a₂+b₁b₂+c₁c₂=0. So every theorem has a dual: swap “point” and “line”, “collinear” and “concurrent,” and it stays true. Three points are collinear exactly when their dual lines are concurrent — and it is the same determinant proving both.
Take three points; their collinearity is det = 0. Dualise each to a line and their concurrency is the determinant of the same matrix — so the two facts are one number: live demo
“Points and lines are different kinds of thing.” — in projective geometry they are interchangeable. Every statement about points collapses to a statement about lines by relabelling, and the algebra does not even change: the determinant that tests collinearity of points is the determinant that tests concurrency of the dual lines. cited
Duality here is not a trick applied to a theorem — it is a symmetry of the plane itself. Swap the words, keep the determinant, and truth is preserved. Gergonne 1826
On the canonical compiler, three points on y=x give collinearity determinant 0, and their dual lines give the identical concurrency determinant 0 — one number, two theorems — while a non-collinear triple gives a non-zero value: