Using the fall equinox to work out the other key celestial dates

Over dinner Austin and I were discussing the first day of Fall that passed this last week. I told him that in the Northern Hemisphere we call this the Fall Equinox. We then worked out that in 3 months we'd get to the first day of winter or the Winter Solstice. And 6 months from that would be the Summer Solstice (Austin was left to work out what month that would be in). I pointed out that September was 1/2 between the Summer Solstice and the Winter Solstice so he worked out that March would be the month for the Spring Equinox.