the conic through five points

Two points determine a line; how many points does it take to determine a conic? The answer is five — give five points (no three of them on a single line) and there is exactly one conic passing through all of them. This is the conic's analogue of 'two points make a line', and it is the foundation for constructing and recognising conics in projective geometry.

Why five? A conic is a homogeneous second-degree equation a x^2 + b y^2 + c z^2 + d x y + e x z + f y z = 0 with six coefficients, but multiplying them all by a constant gives the same curve, so there are really five free ratios to fix. Each point you require the conic to pass through is one linear condition on those coefficients. Five points give five linear equations in five unknowns — generically exactly enough to determine the conic uniquely. There is also a purely synthetic construction, via Pascal's theorem (the 'mystic hexagon'): given five points, treat them as five corners of a hexagon and let a variable line through one of them play the sixth corner; Pascal's collinearity condition then constructs, with a straightedge alone, as many further points of the conic as you like, no equation solved.

This fact is the workhorse of conic geometry — it is how a conic is pinned down from data, how Steiner's projective definition produces a specific curve, and why the conic is, like the line, a fundamentally projective object. Two honest qualifications. First, the 'no three collinear' caveat is essential: if three of the five points lie on a line, the unique 'conic' through them degenerates into a pair of lines rather than a smooth curve, so the genericity matters. Second, five points determine the conic but do not by themselves tell you its affine type — whether the resulting curve reads as an ellipse, parabola, or hyperbola depends on how it meets the line at infinity, which the five finite points alone do not fix until you also fix that line.