Start with a circle and a point P[math][/math]in the plane, and draw lines through P[math][/math]that intersect the circle.[br][br][i]Try repositioning P below, as well as the endpoints of the secant segments.[/i]

Notice that, for fixed P,[math][/math]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 [i]Intersecting (or Crossed) Chords Theorem[/i]. When P lies outside the circle, the equality is known as the [i]Secant-Secant Theorem[/i] or the [i]Secant-Tangent Theorem[/i].

Underlying these theorems, which are often stated separately, is the invariant product demonstrated above, which gives rise to a concept called the [b]power of a point. [/b]The power of a point P with respect to a circle C, denoted [math]\Pi_C\left(P\right)[/math], is equal to the invariant product [math]PA\cdot PB[/math] if P is outside the circle, and equal to the negative invariant product [math]-PA\cdot PB[/math] if P is inside the circle. Importantly, the power of a point can be computed using [i]any[/i] secant, tangent, or chord through P.

Edo period mathematicians seem to have been aware of this concept, and applied it to [url=https://www.geogebra.org/m/vp4rbsjk#material/wvghuu4g]arrangements of multiple circles[/url].