How can I tell if a point belongs to a certain line?

Question

How can I tell if a point belongs to a certain line?

Examples are appreciated, if possible.

Answer 1

As an alternative to the slope/y-intercept method, I chose this approach using Math.Atan2:

// as an extension method
public static bool Intersects(this Vector2 v, LineSegment s) {
    //  check from line segment start perspective
    var reference = Math.Atan2(s.Start.Y - s.End.Y, s.Start.X - s.End.X);
    var aTanTest = Math.Atan2(s.Start.Y - v.Y, s.Start.X - v.X);

    //  check from line segment end perspective
    if (reference == aTanTest) {
        reference = Math.Atan2(s.End.Y - s.Start.Y, s.End.X - s.Start.X);
        aTanTest = Math.Atan2(s.End.Y - v.Y, s.End.X - v.X);
    }

    return reference == aTanTest;
}

The first check reference determines the arcTan from the start point of the line segment to it's end-point. Then from the start point perspective, we determine the arcTan to the vector v.

If those values are equal, we check from the perspective of the end-point.

Simple and handles horizontal, vertical and all else in between.