  The product of the areas of the event horizon and the Cauchy horizon of a
non-extremal black hole equals the square of the area of the horizon of the
black hole obtained from taking the smooth extremal limit. We establish this
result for a large class of black holes using the second order equations of
motion, black hole thermodynamics, and the attractor mechanism for extremal
black holes. This happens even though the area of each horizon generically
depends on the moduli, which are asymptotic values of scalar fields. The
conformal field theory dual to the BTZ black hole facilitates a microscopic
interpretation of the result. In addition, we demonstrate that certain
quantities which vanish in the extremal case are zero when integrated over the
region between the two horizons. We corroborate these conclusions through an
analysis of known solutions.
