Family of Circles through the Intersection of Two Circles

A single equation can describe an entire family of circles sharing the same pair of intersection points. If $S=0$ and $S'=0$ are two circles, then for every value of the parameter $\lambda\neq-1$,

is itself a circle (its $x^2$ and $y^2$ coefficients are both $1+\lambda$), and it passes through every point common to $S=0$ and $S'=0$ -- because such a point satisfies both equations, and hence satisfies $S+\lambda S'=0$ for any $\lambda$ at all. Requiring this family member to also pass through one further specified point, or to satisfy some other single condition, fixes $\lambda$ and selects one particular circle out of the whole family.

The same principle extends to a circle and a line: if $S=0$ is a circle and $L\equiv lx+my+n=0$ is a line meeting it, then

is, for every $k$, a circle through the two points where the line meets the original circle. Taking $L=S-S'$ (the common chord itself) recovers the two-circle family above as a special case, since $S+k(S-S')=(1+k)S-kS'$ is again of the form $\alpha S+\beta S'$.

A closely related, entirely separate construction gives the circle whose diameter is a specified chord directly: if $(x_1,y_1)$ and $(x_2,y_2)$ are the two endpoints of a diameter, the circle is

$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0,$ …