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^2l/n, -a^2m/n). Pole-polar relationships provide elegant solutions to problems about harmonic conjugates and cross-ratios.