Geo related needs

I am working on a website that has a map interface with Openlayers. I
would like to have functionality where I have several points on a map.
If I draw a polygon that encloses some of those points, I would like
to be able to tell which points are within the polygon. That seems
like my main challenge, perhaps there is some gem or an easy way to do
that ? Probably not in SQL though I imagine.

I also want this to be a free app on heroku such as my other apps
are. It’s just a little project of my own. I know that Postrges has a
postgis extension, but it may be a bit complex to install that and I
am not sure if it would be free on heroku. I have installed postgres/
postgis on linux several years ago, but I am using a windows machine
and am not sure how easy that would be either.