Add up the angles between the point in question 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 Function PointInPolygon(ByVal X As Single, ByVal Y _
    As Single, ByVal num_polygon_points As Integer, _
    polygon_points() As POINTAPI) As Boolean
Dim pt As Integer
Dim total_angle As Single

    ' Get the angle between the point and the
    ' first and last vertices.
    total_angle = GetAngle( _
        polygon_points(num_polygon_points).X, _
            polygon_points(num_polygon_points).Y, _
        X, Y, _
        polygon_points(1).X, polygon_points(1).Y)

    ' Add the angles from the point to each other
    ' pair of vertices.
    For pt = 1 To num_polygon_points - 1
        total_angle = total_angle + GetAngle( _
            polygon_points(pt).X, polygon_points(pt).Y, _
            X, Y, _
            polygon_points(pt + 1).X, polygon_points(pt + _
                1).Y)
    Next pt

    ' 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.
    PointInPolygon = (Abs(total_angle) > PI)
End Function