The tangent at point (x1,y1) on the circle is obtained by the T=0 substitution: replace $x^2$ with xx1, $y^2$ with yy1, 2x with x+x1, 2y with y+y1. For x^{2+y}^2=$a^2$, the tangent at (x1,y1) is xx1+yy1=$a^2$. The tangent with slope m to x^{2+y}^2=$a^2$ is y=mx+/-asqrt(1+$m^2$), where the two signs correspond to two parallel tangent lines. The tangent condition for line y=mx+c is $c^2$=$a^2$(1+$m^2$). The normal at any point on a circle passes through the centre — this is a unique property of circles among conics. The chord of contact from external point P(x1,y1) has the same T=0 equation as the tangent. The chord with midpoint (x1,y1) satisfies T=S1. The pair of tangents from an external point satisfies $T^2$=SS1. The angle between tangent pairs is 2*tan^(-1)$\frac{r}{L}$ where L=sqrt(S1). The director circle (locus of perpendicular tangent pairs) is x^{2+y}^2=2$a^2$.