Thermodynamics in the vicinity of a critical endpoint with nonclassical exponents alpha, beta, gamma, delta, ... , is analyzed in terms of density variables (mole fractions, magnetizations, etc.). The shapes of the isothermal binodals or two-phase coexistence curves are found at and near the endpoint for symmetric and nonsymmetric situations. The spectator- (or noncritical-) phase binodal at T=T-e is characterized by an exponent (delta +1)/delta (similar or equal to1.21) with leading corrections of relative order 1/delta (similar or equal to0.21), theta (4)/beta delta (similar or equal to0.34) and 1-(beta delta)(-1) (similar or equal to0.36); in contrast to classical (van der Waals, mean field, etc.) theory, the critical endpoint binodal is singular with a leading exponent (1-alpha)/beta (similar or equal to2.73) and corrections which are elucidated; the remaining, lambda -line binodals also display the "renormalized exponent," (1-alpha)/beta but with more singular corrections. [The numerical values quoted here pertain to (d=3)-dimensional-fluid or Ising-type systems.]