The most important functions are listed below:

// Judge if there exist a parabola passing P[a] and P[b]
// The validation of direction of the parabola is included in the positive sign
bool Valid(int a, int b)
{
    if (P[a].x == P[b].x)
        return false;
    else return (P[a].x - P[b].x) * (P[a].x * P[b].y - P[b].x * P[a].y) > 0;
}

// Judge if the parabola passing P[a] and P[b] also passes P[c]
// Using the determinant for judgement
bool Pass(int a, int b, int c)
{
    return (P[a].x * P[a].x) * P[b].x * P[c].y +
        P[a].x * P[b].y * (P[c].x * P[c].x) +
        P[a].y * (P[b].x * P[b].x) * P[c].x -
        (P[a].x * P[a].x) * P[b].y * P[c].x -
        P[a].x * (P[b].x * P[b].x) * P[c].y -
        P[a].y * P[b].x * (P[c].x * P[c].x) == 0;
}

Note: using long integers (64-bit) to avoid overflow.

Good luck.