How did european leaders discover that a balance of power did not decrease the chances for war among them?

The balance of power was an 18th century effort made to keep the peace after the Napoleonic wars. The Congress of Vienna, which sought to establish stability after those wars, aimed at achieving a balance of strength between the five main powers of Europe: Britain, Russia, Prussia, Austria, and France. (France was still to be considered a great power, even after Napoleon was defeated.)

The problem with the balance of power is that ultimately it became a race for power. In the late 19th and early 20th century, the powerful nations of Europe began to ramp up their military strength over against one another. They entered into alliances of one group vs. another in the event of war. They competed for control of other territories overseas, pursuing imperialistic ambitions. Ultimately, all this competition for power (which is what the "balance" of power became) led to the outbreak of The Great War, which we know today as World War I.