Find the curvature of a curve with a kissing circle using Processing

To try to programatically solve this problem, I first set up a p5.js environment where I can plot a curve.

fr = 10
currentPoint=15
function setup() {
  createCanvas(400, 400);
  background(220);
  frameRate(fr);
}
isDrawing = false;
points = [];
var e;
function draw() {
  if(isDrawing){
    points.push([mouseX, mouseY, points.length]);
    ellipse(mouseX, mouseY, 2, 2);
    if(points.length > 1){
      line(mouseX, mouseY, points[points.length - 2][0], points[points.length - 2][1]);
    }
  }
  if(points.length === currentPoint){
    fill(255,123,0);
    ellipse(points[currentPoint-2][0], points[currentPoint-2][1], 5, 5)
    fill(255,0,0);
    ellipse(points[currentPoint-1][0], points[currentPoint-1][1], 5, 5)
    fill(255,123,0);
    ellipse(points[currentPoint-3][0], points[currentPoint-3][1], 5, 5)
  }
}
function mousePressed() {
  if (isDrawing) {
    isDrawing = false;
    console.log(points)
  } else {
    isDrawing = true;
  }
}
function keyPressed() {
  if (keyCode === UP_ARROW) {
    currentPoint += 1;
  } else if (keyCode === DOWN_ARROW) {
    currentPoint -= 1;
  }

  console.log('Current point: ', currentPoint);
}

Here, I have highlighted three points in red, which are the points I will use to create the kissing circle. Assuming that the three points do not lie on a straight line, we can find an unique circle which passes through them. We can use that circle, and its radius to approximate the curvature of the curve at the point. I came across a stack exchange post which was basically tacking the same problem. However, to try to create a system which solves the system of equations which are provided here seemed rather difficult, given that they are non-linear. I tried to analytically solve the equation for r to create a function r(p1, p2, p3). However, I was not able to do this by hand. (blah blah, I gave up on solving this.)

I then went back to thinking about ways in which I could find the radius, given the three points. I figured that I could create my own system of equations, such that it is easier to find the radius. Since all points lie on the circle, we could use the equation for a circle to create a set of equations which represents this circle. This was definitely a step in the right direction, but this is still non-linear, so its difficult to solve. I could of course use numerical methods like Monte Carlo to approximate the values, but I figured it shouldn’t be too hard to find a way to turn this into a linear system of equations.

I was able to reduce it to a linear equation by creating three new variables A, B and C, which I can use to solve for x_m, r and y_m later on.

Finally, I believe I can use these matrices to create a program to find a circle, and a curvature in the end.