The polar of point P(x1,y1) with respect to circle x^{2+y}^2=$a^2$ is xx1+yy1=$a^2$ (the same T=0 formula). When P is on the circle, the polar is the tangent. When P is outside, the polar is the chord of contact. When P is inside, the polar is the line whose pole is P. The pole and polar are conjugate: if the polar of P passes through Q, then the polar of Q passes through P. This duality is useful in geometric proofs. The pole of a line lx+my+n=0 is (-$a^2$*l/n, -$a^2$*m/n). Pole-polar relationships provide elegant solutions to problems about harmonic conjugates and cross-ratios.