Eurystheus was king of Tiryns.

King Eurystheus and Hercules were related because....

Hercules was King Erytheus Eurystheus nephew.

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Eurystheus doesn't fight the monsters because he's afraid of them; Eurystheus is described in many renditions as "cowardly"

Eurystheus was one of the grandsons of Perseus (the original epic hero) and was Hera's candidate for the hero that would rid the world of the remaining Cthonic monsters. Zeus' candidate was Heracles.

In the end, Heracles won, and it was Eurystheus that gave Heracles his twelve labors (originally 10, but he ruled that the Hydra didn't count because he received help and the stables didn't count because he received payment).

Yes. Eurystheus and Hercules are 2nd cousins once removed.