Solving problems by deciding a logic
I think the most spectacular example of this approach is Thomas Hales' proof of the Kepler conjecture (aka Hilbert's 18th problem). 

In the 1950s, Fejes Toth proved that the Kepler conjecture was equivalent to a large collection of formulas in the first-order theory of real closed fields. Since Tarski had shown this theory to be decidable, he went on to speculate that if a sufficiently fast computer could be built, the Kepler conjecture could be resolved by machine. 

Half a century later, Thomas Hales actually made this strategy work (after proving a lot of difficult theorems to further simplify the set of formulas the computer needed to check)!