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I am not sure by what you mean by the energy of an isolated system being constant at equilibrium specifically, but the definition of an isolated system is a system that cannot exchange either matter or energy with its surroundings. There can be energy transformations within an isolated system, but there is not change in the total energy level. Because there is no change in the total energy level of an isolated system, we say that energy in an isolated system is conserved. This energy conservations is stated in the first law of thermodynamics.
A reaction has reached equilibrium when the reaction has stopped progressing, so that the amount of reactants that have turned into products remains constant, and the amount of reactants left over stays constant. The equilibrium constant is the ratio of products to reactants when the reaction has reached equilibrium.
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