pole and polar
A conic does more than sit in the plane — it sets up a perfect dictionary between points and lines. To every point it assigns a particular line, and to every line a particular point, in a way that is its own inverse. The point is called the pole, the line its polar, and this pole-polar correspondence is the cleanest concrete realisation of duality: the conic itself turns 'point' into 'line' and back.
Here is the construction for a point P outside a conic. Draw the two tangent lines from P that touch the conic; the chord joining the two points of tangency is the polar of P. For a point P inside, the polar is found instead by drawing any two secants through P: each secant meets the conic in two points, and on each secant the two intersection points together with P determine a harmonic conjugate point; the polars are tied to those harmonic conjugates, and the locus of fourth harmonic points is the polar line. The relationship is reciprocal (a true polarity): if the polar of P passes through Q, then the polar of Q passes through P — that symmetry is the conjugate-point relation. In coordinates it is beautifully simple: for the conic with symmetric matrix S, the polar of point p is the line S p, and a point lies on its own polar exactly when it lies on the conic, in which case the polar is the tangent there.
Pole-polar duality matters because it gives duality a concrete engine — it converts theorems about points on a conic into theorems about tangent lines, and it is the right tool for constructions involving conics with a straightedge alone (the harmonic-conjugate machinery does all the work, no measuring needed). Two honest notes. First, the whole relationship is defined relative to a chosen conic; 'the polar of P' is meaningless until you say polar with respect to which conic. Second, a point on the conic is special: it is its own pole's partner in the sense that its polar is precisely the tangent line at that point, the limiting case where the two tangents from P merge.
Take the unit circle x^2 + y^2 = 1 and the external point P = (2, 0). Its polar is the vertical line x = 1/2 (in general the polar of (x_0, y_0) for this circle is x_0 x + y_0 y = 1). Check the reciprocity: the polar passes through, say, (1/2, h); the polar of (1/2, h) is (1/2)x + h y = 1, which indeed passes through P = (2, 0) since (1/2)(2) + h(0) = 1.
For a point outside: its polar is the chord joining the two points where the tangents touch.
Pole and polar are always taken with respect to a specific conic — there is no absolute polar. A point lies on the conic exactly when it lies on its own polar, and then its polar is the tangent at that point. The relation is reciprocal: P on the polar of Q is the same as Q on the polar of P (conjugate points).