a simple demo to testing the formula: http://en.wikipedia.org/wiki/Line-line_intersection

find_the_intersection_of_two_lines.pde

float[] x = {
  88, 431, 431, 124
};
float[] y = {
  115, 367, 188, 331
};

void setup() {
  size(640, 480);
}

void draw() {
  background(200);
  
  line(x[0], y[0], x[1], y[1]);
  line(x[2], y[2], x[3], y[3]);
  
  float X,Y,delta;
  delta = (x[0]-x[1])*(y[2]-y[3])-(y[0]-y[1])*(x[2]-x[3]);
  
  float x1y2y1x2 = (x[0]*y[1]-y[0]*x[1]);
  float x3y4y3x4 = (x[2]*y[3]-y[2]*x[3]);
  X = (x1y2y1x2*(x[2]-x[3])-(x[0]-x[1])*x3y4y3x4)/delta;
  Y = (x1y2y1x2*(y[2]-y[3])-(y[0]-y[1])*x3y4y3x4)/delta;
  
  ellipse(X,Y,10,10); // intersection
  
  x[3] = mouseX;
  y[3] = mouseY;
  
}

void mousePressed() {
  println(mouseX+" "+mouseY);
}