Find circumcircle of a triangle

Find circumcircle of a triangle

Demo

Click on the panel below to see the demo

Steps of the algorithm

We all know that the circumcircle of a triangle $A B C$ is the intersection of the three perpendicular bisectors of it. Based on this property, we can calculate the coordinates of the center of the circumcircle of triangle $A B C$ according to the following steps:

Find the midpoints $M_{1}$ and $M_{2}$ of sides $A B$ and $A C$ respectively
Calculate the equation of the line perpendicular to $A B$ and $A C$ at the midpoints $M_{1}$ and $M_{2}$ just found
Find the intersection of these two perpendicular lines.

Find the mid point

Given two points $A (x_{A}, y_{A}), B (x_{B}, y_{B})$ , the mid point of the line segment $A B$ is calculated as $M (x_{M}, y_{M}) = M (\frac{x_{A} + x_{B}}{2}, \frac{y_{A} + y_{B}}{2})$

Find the perpendicular vectors to a line

Given a line defined by 2 points: $A (x_{A}, y_{A}), B (x_{B}, y_{B})$ . There are 2 perpendicular vectors to the line: $v_{1} ⊥ \vec{A B}, v_{2} ⊥ \vec{A B}$ with $\vec{A B} = (x_{B} - x_{A}, y_{B} - y_{A})$ then we have $v 1 = (y_{B} - y_{A}, x_{B} - x_{A}), v_{2} = (y_{A} - y_{B}, x_{A} - x_{B})$

Code

				
					// find perpendicular vector of P1 P2
function perpendicularOfALine(p1, p2){    
    const v = createVector(p2.x - p1.x, p2.y - p1.y).normalize();
    const pVector1 = createVector(v.y, -v.x);    
    const pVector2 = createVector(-v.y, v.x);

    return [pVector1, pVector2];
}

Find intersection of 2 lines

Given 2 points $A, B$ we calculate the formula of the line go through those points in form $a x + b y + c = 0$ by:

$(d) : a x + b y + c = 0 ⟹ {\begin{cases} a = y_{B} - y_{A} \\ b = x_{A} - x_{B} \\ c = - a \times x_{A} - b \times y_{B} \end{cases}$

We have the formula to calculate the intersection of two lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ is as follow:

$(x, y) = (\frac{b_{1} \times c_{2} - b_{2} \times c_{1}}{a_{1} \times b_{2} - a_{2} \times b_{1}}, \frac{c_{1} \times a_{2} - c_{2} \times a_{1}}{a_{1} \times b_{2} - a_{2} \times b_{1}})$

Code

				
					// find intersection of 2 lines
function findIntersectionOfTwoLines(p1, p2, q1, q2){
    const a1 = p2.y - p1.y;
    const b1 = p1.x - p2.x;
    const c1 = -(a1 * p1.x + b1 * p1.y);

    const a2 = q2.y - q1.y;
    const b2 = q1.x - q2.x;
    const c2 = -(a2 * q1.x + b2 * q1.y);

    const det = a1 * b2 - a2 * b1;
    if (det == 0){
        return createVector(0, 0);
    } else {
        const x = (b1 * c2 - b2 * c1) / det;
        const y = (a2 * c1 - a1 * c2) / det;
        return createVector(x, y);
    }    
}