Practice and the Chapter IV test

Draw the diameter through the point P at distance p from the centre O of a circle of radius R. If P is inside, the diameter splits at P into the parts R − p and R + p, so for any chord AB through P we get AP·PB = (R − p)(R + p) = R² − p². If P is outside, the secant through the centre has PC = p − R and PD = p + R, so for every secant through P we get PA·PB = p² − R². For the tangent, PA² = p² − R², that is PA = √(p² − R²); this also follows from the Pythagorean theorem, as the radius is perpendicular to the tangent. These formulas let us compute an unknown segment directly.