Define ellipse given 3 of its points

Long story short, I have a camera sensor that detects 3 points a reports them as x,y pairs. All x and y values are positive.

It's been a long time since I've done geometry and algebra and I'm struggling with getting started. Maybe someone can help me.

I know the points are on an ellipse, and I know the major axis is vertical. I don't know the foci, but don't need them in the end.

Ultimately, I need to find the center of the ellipse.

Can this be done with algebra?

Thanks for any help!