Consider two circles in the plane. Let E be their external point of similarity.[br][br][i]You can drag the circles below and change the endpoints of the secant segments.[/i]

Notice that, for a given arrangement of the circles, there are four products of segment lengths with the same value. This number is known as the [b]common power of the two circles. [/b]It can also be written in terms of the [url=https://www.geogebra.org/m/vp4rbsjk#material/nzbwvznc]power of E[/url] with respect to circles 1 and 2 as [math]\sqrt{\Pi_1\left(E\right)\cdot\Pi_2\left(E\right)}[/math].

Such an arrangement generally gives rise to two (additional) cyclic quadrilaterals.

You also have the option of displaying the [b]circle of inversion[/b] with respect to which the original two circles are inverses of each other. More specifically, point A inverts to point D, B inverts to C, etc. The circle of inversion has center E and radius equal to the square root of the common power.

The theory of circle inversion, as it is traditionally known in the West, was developed by European mathematicians starting in the 1820s. The extent to which Edo period Japanese mathematicians had a similar theory is a matter of debate among scholars. There is little doubt, however, that they understood related concepts like the power of a point and the common power of two circles.