Secants, Chords, and Tangents
Start with a circle and a point Pin the plane, and draw lines through Pthat intersect the circle.
Try repositioning P below, as well as the endpoints of the secant segments.
Notice that, for fixed P,there are products of segment lengths that do not depend on the choice of intersecting points B or D. When P lies inside the circle, the relevant equality of products is known as the Intersecting (or Crossed) Chords Theorem. When P lies outside the circle, the equality is known as the Secant-Secant Theorem or the Secant-Tangent Theorem.
Underlying these theorems, which are often stated separately, is the invariant product demonstrated above, which gives rise to a concept called the power of a point. The power of a point P with respect to a circle C, denoted , is equal to the invariant product if P is outside the circle, and equal to the negative invariant product if P is inside the circle. Importantly, the power of a point can be computed using any secant, tangent, or chord through P.
Edo period mathematicians seem to have been aware of this concept, and applied it to arrangements of multiple circles.