Balance and strength on one leg is essential for successful running.
Consider three truths:
1) Running is a series of hops from one foot to the other. Upon landing, you perform a partial, one-leg squat in preparation for the next hop.
2) Research suggests that strength training aids running performance, and
3) The principle of specificity says that to improve at a given physical task, training should resemble that task.
Given these truths, it seems clear you can benefit by including single-leg squats as part of a regular strength program. The hop-and-land sequence of running demands strength and stability in order to perform well and avoid injury.
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