The well known property of quadratic functions, that the tangents at either end of a chord of a parabola meet in a point aligned vertically with the midpoint of the chord is extended to polynomials of degree d. Given two distinct points on a polynomial of degree d, the Taylor polynomials of degree d - 1 at those points meet in d - 1 points whose mean is vertically aligned with the midpoint of the chord joining the two points. This characterizes polynomials of degree d.