One way to determine whether a point lies within a polygon is to add up the angles between the point and adjacent points on the polygon taken in order. If the total of all the angles is 2 * PI or -2 * PI, then the point is inside the polygon. If the total is zero, the point is outside. You can verify this intuitively with some simple examples using squares or triangles.

// Return True if the point is in the polygon.
public bool PointInPolygon(float X, float Y)
{
    // Get the angle between the point and the
    // first and last vertices.
    int max_point = Points.Length - 1;
    float total_angle = GetAngle(
        Points[max_point].X, Points[max_point].Y,
        X, Y,
        Points[0].X, Points[0].Y);

    // Add the angles from the point
    // to each other pair of vertices.
    for (int i = 0; i < max_point; i++)
    {
        total_angle += GetAngle(
            Points[i].X, Points[i].Y,
            X, Y,
            Points[i + 1].X, Points[i + 1].Y);
    }

    // The total angle should be 2 * PI or -2 * PI if
    // the point is in the polygon and close to zero
    // if the point is outside the polygon.
    return (Math.Abs(total_angle) > 0.000001);
}