No way to do it with the line() and strokeWeight command. What about drawing a triangle instead ?  http://www.processing.org/reference/triangle_.html cause thats more or less what it is. 

noted with thanks!

currently I'm drawing a group of points with changing strokeweight and tranparency to realize this effect :

yes, as said, that could be a solution as well. if you dont want a straight line. you might also use the bezierPoint() Function with the same approach. although the transparency doesnt work well because of the overlapping circles.

thanks!

small tweak:

1. void setup() {
     size(500, 500);
     smooth();
   }
   
   void draw() {
     background(0, 0, 255);
     noStroke();
     noFill();
     bezier(85, 120, 110, 210, 390, 590, 315, 280);
   
     int steps =300;
     for (int i = 0; i <= steps; i++) {
       fill(255, 0, 0, map(steps-i, 0, steps, 0, 255));
       float t = i / float(steps);
       float x = bezierPoint(85, 110, 390, 315, t);
       float y = bezierPoint(120, 210, 590, 280, t);
       ellipse(x, y, i/10.0, i/10.0);
     }
   }