The momentum distribution of the energy gap opening at the Fermi level of superconductors is a direct fingerprint of the pairing mechanism. While the phase diagram of the iron-based superconductors promotes antiferromagnetic fluctuations as a natural candidate for electron pairing, the precise origin of the interaction is highly debated. We used angle-resolved photoemission spectroscopy to reveal directly the momentum distribution of the superconducting gap in FeTe1-xSex, which has the simplest structure of all iron-based superconductors. We found isotropic superconducting gaps on all Fermi surfaces whose sizes can be fitted by a single gap function derived from a strong coupling approach, strongly suggesting local antiferromagnetic exchange interactions as the pairing origin.