the power of a point
Pick any fixed point and any circle, and draw a straight line through the point that meets the circle. It will cross the circle at two spots (or, if the line just grazes, at one). Multiply the two distances from your point to those crossing spots, and a small miracle occurs: you get the same product no matter which line through the point you chose. That fixed product is the power of the point with respect to the circle.
Precisely, the power of a point P relative to a circle of centre O and radius r is the number |OP|^2 - r^2, and it equals the signed product of the distances from P to the two intersection points of any line through P with the circle. This single idea unifies three school theorems. If two chords cross inside the circle at P, splitting one into pieces a, b and the other into c, d, then a·b = c·d (the chord-chord, or intersecting-chords, theorem). If two secants are drawn from an external point P, then (whole first secant)·(its external part) = (whole second)·(its external part). And if a secant and a tangent come from an external point, the tangent length squared equals the secant's whole-by-external product: t^2 = (whole secant)·(external part) — the tangent-secant theorem. For an external point all products are positive (the power is positive); for a point inside, the power is negative, which is why the inside (chord-chord) version is stated with both pieces measured as positive lengths.
The power of a point is one of the great organizing ideas of circle geometry: it proves the three product relations at a stroke, characterizes when four points are concyclic (equal powers / the radical axis), and underlies the radical center of three circles. The caveats worth flagging: the products match only for lines through the same point and the same circle; the tangent case is the limiting secant where the two intersection points merge, so 'whole times external' becomes 'tangent squared'; and the sign convention matters — the power is positive outside, zero exactly on the circle, and negative inside, so quoting it as a bare 'product' without minding inside-versus-outside can flip a sign.
Two chords cross at P inside a circle, splitting one into 3 and 8 and the other into 4 and x. Chord-chord gives 3·8 = 4·x, so x = 6. From an external point with a tangent of length t and a secant whose whole length is 9 and external part 4, the tangent-secant rule gives t^2 = 9·4 = 36, so t = 6.
Chord-chord a·b = c·d and tangent-secant t^2 = (whole)·(external), both from one power.
The products agree only for lines through the same point and the same circle. The power is positive outside the circle, zero on it, negative inside — track the sign, and read the tangent case as the limit where two intersection points merge.