The equation of a pair of tangents from $(x_1,y_1)$ to the circle $x^2+y^2+2gx+2fy+c=0$ is given by $T^2= SS_1$ where:
$S= x^2+y^2+2gx+2fy+c \\ S_1= x_1^2+y_1^2+2gx_1+2fy_1+c \\ T = xx_1+yy_1+g(x+x_1)+f(y+y_1)+c $
Can someone provide me the non calculus proof for this?