An Affine Reflection Across a Curve and a Geometric Proof of Young's Integral Identity

I introduce a new geometric transformation , defined by two simple axioms: the midpoint of a point and its image lies on a given mirror curve, and the segment joining them has slope equal to the negative of the mirror's slope at that midpoint. For a linear mirror, the reflected curve is given by the composition I = m ∘ f⁻¹ ∘ m, and I prove that the net signed area between a function and the mirror is equal to the area between the mirror and its reflection. This area-preserving property yields a purely geometric proof of Young's integral identity for inverse functions, including the general form with an arbitrary secant line. For a linear mirror, the reflection is an affine involution with determinant -1, providing an alternative, purely geometric verification of the area theorem. When the mirror is curved, the area balance breaks, and I derive an exact Jacobian that shows how the defect is controlled by the mirror's curvature. The framework unifies classical inequalities with affine geometry and invites further study of singularities and higher-dimensional analogues.