Orthogonal Circles

Two circles are orthogonal when they intersect at right angles, i.e. when the angle $\theta$ of the previous section equals $90^\circ$. Since $\cos90^\circ=0$, setting the numerator of the angle formula to zero gives the orthogonality condition:

for circles $x^2+y^2+2gx+2fy+c=0$ and $x^2+y^2+2g'x+2f'y+c'=0$. Equivalently, from the cosine-rule identity, orthogonality is exactly the Pythagorean relation $d^2=r_1^2+r_2^2$ between the centre-distance and the two radii.

Orthogonality has a clean geometric picture: at either point of intersection of two orthogonal circles, the tangent to one circle passes straight through the centre of the other. This is because a tangent from a circle's centre to another circle, of length equal to that circle's own radius, is only possible when the Pythagorean relation above holds -- exactly the orthogonality condition. …