How to find a middle point of a beizer curve?

So, the formula to translate the 4 points in a function over time is the following (image from wikipedia):

Since you want the middle, and t ranges from 0 to 1, you just have to set t = 1/2

So

B(1/2) = 1/8 P0 + 3/8 P1 + 3/8 P2 + 1/8 P3

You can use the bezierPoint() function that comes with P5.js.

From the reference:

noFill();
var x1 = 85,
 x2 = 10,
 x3 = 90,
 x4 = 15;
var y1 = 20,
 y2 = 10,
 y3 = 90,
 y4 = 80;
bezier(x1, y1, x2, y2, x3, y3, x4, y4);
fill(255);
var steps = 10;
for (var i = 0; i <= steps; i++) {
  var t = i / steps;
  var x = bezierPoint(x1, x2, x3, x4, t);
  var y = bezierPoint(y1, y2, y3, y4, t);
  ellipse(x, y, 5, 5);
}

You'd probably want to use a value of 0.5 for t to get the midpoint.