Some antipodal distance-regular graphs of diameter three arise as the graph induced by the vertices at distance two from a given vertex in a strongly regular graph. We show that if every vertex in a strongly regular graph G has this property, then G is the noncollinearity graph of a special type of semipartial geometry. As these semipartial geometries have all been classified, we obtain a list of the antipodal distance-regular graphs of diameter three that can arise in this way.