Find the corner points of a set of pixels that make up a quadrilateral boundary

I have a situation where I have a set of pixels that make up the border of a quadrilateral (very close to square). I'm trying to determine the location of the corners as best as possible and have been struggling for a while now. My first thought was to determine the straight lines of the border and then calculate the corner points, but I don't have access to OpenCV or other image processing libraries, unfortunately.

Below are three cases where the black outline is the image boundary and the red outline is the quadrilateral boundary. I have a list of all of the pixels that make up the red boundary and the red boundary thickness may vary.

My initial thought was that I could just find the pixel that is closest to each of the four image boundaries, however this won't quite work for the first case where the inner quadrilateral isn't tilted.

Any thoughts on how to tackle this problem would be great. I'm coding in dart, but am looking for a psuedocode answer that I can implement myself.

(I have seen this post, which is similar to my problem, but I think there should be a simpler solution for my problem since I have access to all of the boundary points of the quadrilateral)

Having a list of all rectangle boundary pixels, you can use simple methods like this:

Calculate gravity center of rectangle (just sum X- and Y- coordinates of pixels and divide by their number) - it is diagonal intersection.

Find the farthest pixels - they are corners.

In case of bad quality of data set (empty places, excessive pixels) center calculation might be inexact. So you can apply Hough transform to extract sides (as lines) and calculate their intersections.