C#-Code

using System;
using static System.Console;
using System.Text;

Double[][] XY_Test = new Double[][]  {
    new double[] {  0,   0 }, // P1 (x1, y1)
    new double[] {  1,   1 }, // P2 (x2, y2)
    new double[] {  1,   1 }, // P3 (x3, y3)
    new double[] {-10,   0 }  // P4 (x4, y4)
};

WriteLine(Environment.NewLine + Intersection(XY_Test));
ReadKey();

static string ToMyString(Double[][] xy)  {
    StringBuilder sb = new();
    for (var i = 0; i < xy.Length; i++)  {
        for (var j = 0; j < xy[i].Length; j++)
            sb.Append($"{xy[i][j].ToString("F2"),6}");
        sb.AppendLine();
    }
    return sb.ToString();
}

static bool Intersection(Double[][] xy)  {

    WriteLine(ToMyString(xy));

    Double dx0 = xy[1][0] - xy[0][0]; WriteLine($"dx0  delta x L12  {dx0}"); //delta x L12
    Double dy0 = xy[1][1] - xy[0][1]; WriteLine($"dy0  delta y L12  {dy0}"); //delta y L12

    Double dx1 = xy[3][0] - xy[2][0]; WriteLine($"dx1  delta x L34  {dx1}"); //delta x L34   
    Double dy1 = xy[3][1] - xy[2][1]; WriteLine($"dy1  delta y L34  {dy1}"); //delta y L34

    Double DX = xy[2][0] - xy[0][0]; 
    Double DY = xy[2][1] - xy[0][1]; 

    Double det = dy0 * dx1 - dy1 * dx0;  

    if (det == 0)
        return false;

    Double det1 = DY * dx1 - dy1 * DX;
    Double det2 = dy0 * DY - DX * dx0;

    WriteLine($"det {det}    det1 {det1}    det2 {det2}");

    Double t1 = det1 / det;
    Double t2 = det2 / det;

    WriteLine($"t1 {t1.ToString("F2"),6} t2 {t2.ToString("F2"),6}");

    return (t1 <= 1) && (t2 <= 1) && (t1 >= 0) && (t2 >= 0);
}