Centre and radius of a circle passing through 3 points in a plane

Centre and radius of a circle passing through 3 points in a plane
You are encouraged to solve this task according to the task description, using any language you may know.

Write a function which returns the centre and radius of a circle passing through three point in a plane. Demonstrate the function using the points (22.83,2.07) (14.39,30.24) and (33.65,17.31)

ALGOL 68

Follows the lines of the C++ code at geeksforgeeks.org.

BEGIN # find the centre and radius of a circle through 3 points                                     #
      # follows the lines of the C++ code at                                                        #
      # https://www.geeksforgeeks.org/equation-of-circle-when-three-points-on-the-circle-are-given/ # 

    MODE POINT  = STRUCT( REAL x, y );
    MODE CIRCLE = STRUCT( POINT centre, REAL radius );

    # returns the circle that passes through p1, p2 and p3                                          #
    PROC find circle = ( POINT p1, p2, p3 )CIRCLE:
         BEGIN
            REAL x1 = x OF p1, y1 = y OF p1, x2 = x OF p2, y2 = y OF p2, x3 = x OF p3, y3 = y OF p3;
            REAL x12  = x1 - x2
               , x13  = x1 - x3
               , y12  = y1 - y2
               , y13  = y1 - y3
               , y31  = y3 - y1
               , y21  = y2 - y1
               , x31  = x3 - x1
               , x21  = x2 - x1
               ;
            REAL sx13 = x1^2 - x3^2
               , sy13 = y1^2 - y3^2
               , sx21 = x2^2 - x1^2
               , sy21 = y2^2 - y1^2
               ;
            REAL f = ( ( ( sx13 + sy13 ) * x12 )
                     + ( ( sx21 + sy21 ) * x13 )
                     )
                   / ( 2 * ( ( y31 * x12 ) - ( y21 * x13 ) ) )
               , g = ( ( ( sx13 + sy13 ) * y12 )
                     + ( ( sx21 + sy21 ) * y13 )
                     )
                   / ( 2 * ( ( x31 * y12 ) - ( x21 * y13 ) ) )
               ;
            REAL c = - (x1^2) - (y1^2) - ( 2 * g * x1 ) - ( 2 * f * y1 );
 
            ( ( -g, -f ), sqrt( g^2 + f^2 - c ) )
         END # find circle # ;

    CIRCLE c = find circle( ( 22.83, 2.07 ), ( 14.39, 30.24 ), ( 33.65, 17.31 ) );

    print( ( "Centre = ( ",   fixed( x OF centre OF c, -10, 6 ) ) );
    print( ( ", ",            fixed( y OF centre OF c, -10, 6 ) ) );
    print( ( " ), Radius = ", fixed(      radius OF c, -10, 6 ), newline ) )

END

Output:

Centre = (  18.978516,  16.265411 ), Radius =  14.708624

BASIC

BASIC256

Translation of: FreeBASIC

call findCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31)
end

subroutine findCircle(x1, y1, x2, y2, x3, y3)
	x12 = x1 - x2
	x13 = x1 - x3
	y12 = y1 - y2
	y13 = y1 - y3
	y31 = y3 - y1
	y21 = y2 - y1
	x31 = x3 - x1
	x21 = x2 - x1

	sx13 = x1 * x1 - x3 * x3
	sy13 = y1 * y1 - y3 * y3
	sx21 = x2 * x2 - x1 * x1
	sy21 = y2 * y2 - y1 * y1

	f = (sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13) / (y31 * x12 - y21 * x13) / 2
	g = (sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13) / (x31 * y12 - x21 * y13) / 2

	c = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1
	h = -g
	k = -f
	r = sqr(h * h + k * k - c)

	print "Centre is at ("; h; ", "; k; ")"
	print "Radius is "; r
	print
	print "Check radius as the distance between the centre and the first point:"
	print sqr((22.83 - h) ^ 2 + (2.07 - k) ^ 2)
end subroutine

Output:

Centre is at (18.9785156601, 16.2654107977)
Radius is 14.708624

Check radius as the distance between the centre and the first point:
14.708624

Chipmunk Basic

Translation of: FreeBASIC

Works with: Chipmunk Basic version 3.6.4

100 sub findcircle(x1,y1,x2,y2,x3,y3)
110 x12 = x1-x2
120 x13 = x1-x3
130 y12 = y1-y2
140 y13 = y1-y3
150 y31 = y3-y1
160 y21 = y2-y1
170 x31 = x3-x1
180 x21 = x2-x1
190 '
200 sx13 = x1*x1-x3*x3
210 sy13 = y1*y1-y3*y3
220 sx21 = x2*x2-x1*x1
230 sy21 = y2*y2-y1*y1
240 '
250 f = (sx13*x12+sy13*x12+sx21*x13+sy21*x13)/(y31*x12-y21*x13)/2
260 g = (sx13*y12+sy13*y12+sx21*y13+sy21*y13)/(x31*y12-x21*y13)/2
270 '
280 c = -x1*x1-y1*y1-2*g*x1-2*f*y1
290 h = -g
300 k = -f
310 r = sqr(h*h+k*k-c)
320 '
330 print "Centre is at ( ";h;", ";k;")"
340 print "Radius is ";r
350 print
360 print "Check radius as the distance between the centre and the first point:"
370 print sqr((22.83-h)^2+(2.07-k)^2)
380 end sub
390 cls
400 findcircle(22.83,2.07,14.39,30.24,33.65,17.31)
410 end

Gambas

Translation of: FreeBASIC

Sub findCircle(x1 As Float, y1 As Float, x2 As Float, y2 As Float, x3 As Float, y3 As Float)  

  Dim x12 As Float = x1 - x2  
  Dim x13 As Float = x1 - x3  
  Dim y12 As Float = y1 - y2  
  Dim y13 As Float = y1 - y3  
  Dim y31 As Float = y3 - y1  
  Dim y21 As Float = y2 - y1  
  Dim x31 As Float = x3 - x1  
  Dim x21 As Float = x2 - x1  
  
  Dim sx13 As Float = x1 * x1 - x3 * x3  
  Dim sy13 As Float = y1 * y1 - y3 * y3  
  Dim sx21 As Float = x2 * x2 - x1 * x1  
  Dim sy21 As Float = y2 * y2 - y1 * y1  
  
  Dim f As Float = (sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13) / (y31 * x12 - y21 * x13) / 2  
  Dim g As Float = (sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13) / (x31 * y12 - x21 * y13) / 2  
  
  Dim c As Float = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1  
  Dim h As Float = -g  
  Dim k As Float = -f  
  Dim r As Float = Sqr(h * h + k * k - c)  
  
  Print "Centre is at (" & h & ", " & k & ")"  
  Print "Radius is "; r  
  Print  
  Print "Check radius as the distance between the centre and the first point:"  
  Print Sqr((22.83 - h) ^ 2 + (2.07 - k) ^ 2)  

End Sub  

Public Sub Main()  
  
  findCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31) 
  
End

Output:

Centre is at (18.9785156601488, 16.2654107977159)
Radius is 14,7086239783342

Check radius as the distance between the centre and the first point:
14,7086239783342

PureBasic

Procedure.d findCircle(x1, y1, x2, y2, x3, y3)
  Define.d x12, x13, y12, y13, y31, y21, x31, x21, sx13, sy13, sx21, sy21
  Define.d f, g, c, h, k, r
  
  x12 = x1 - x2
  x13 = x1 - x3
  y12 = y1 - y2
  y13 = y1 - y3
  y31 = y3 - y1
  y21 = y2 - y1
  x31 = x3 - x1
  x21 = x2 - x1
  
  sx13 = x1 * x1 - x3 * x3
  sy13 = y1 * y1 - y3 * y3
  sx21 = x2 * x2 - x1 * x1
  sy21 = y2 * y2 - y1 * y1
  
  f = (sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13) / (y31 * x12 - y21 * x13) / 2
  g = (sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13) / (x31 * y12 - x21 * y13) / 2
  
  c = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1
  h = -g
  k = -f
  r = Sqr(h * h + k * k - c)
  
  PrintN("Centre is at (" + Str(h) + ", " + Str(k) + ")")
  PrintN("Radius is " + Str(r))
  PrintN("")
  PrintN("Check radius as the distance between the centre and the first point: " + Str(Sqr(Pow((22.83 - h), 2) + Pow((2.07 - k), 2))))
EndProcedure

OpenConsole()
findCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31)
PrintN(#CRLF$ + "--- terminado, pulsa RETURN---"): Input()
PrintN(#CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()

Yabasic

Translation of: FreeBASIC

findCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31)
END

SUB findCircle(x1, y1, x2, y2, x3, y3)
	x12 = x1 - x2
	x13 = x1 - x3
	y12 = y1 - y2
	y13 = y1 - y3
	y31 = y3 - y1
	y21 = y2 - y1
	x31 = x3 - x1
	x21 = x2 - x1

	sx13 = x1 * x1 - x3 * x3
	sy13 = y1 * y1 - y3 * y3
	sx21 = x2 * x2 - x1 * x1
	sy21 = y2 * y2 - y1 * y1

	f = (sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13) / (y31 * x12 - y21 * x13) / 2
	g = (sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13) / (x31 * y12 - x21 * y13) / 2

	c = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1
	h = -g
	k = -f
	r = SQRT(h * h + k * k - c)

	PRINT "Centre is at (", h, ", ", k, ")"
	PRINT "Radius is ", r
	PRINT
	PRINT "Check radius as the distance between the centre and the first point:"
	PRINT SQRT((22.83 - h) ^ 2 + (2.07 - k) ^ 2)
END SUB

Output:

Centre is at (18.9785, 16.2654)
Radius is 14.7086

Check radius as the distance between the centre and the first point:
14.7086

C++

This follows the lines of the C++ code here.

#include <iostream>
#include <math.h>
using namespace std;
 
void findCircle(float x1, float y1, float x2, float y2, float x3, float y3) {
    float x12 = x1 - x2;
    float x13 = x1 - x3; 
    float y12 = y1 - y2;
    float y13 = y1 - y3;
    float y31 = y3 - y1;
    float y21 = y2 - y1;
    float x31 = x3 - x1;
    float x21 = x2 - x1;
    
    float sx13 = pow(x1, 2) - pow(x3, 2);
    float sy13 = pow(y1, 2) - pow(y3, 2);
    float sx21 = pow(x2, 2) - pow(x1, 2);
    float sy21 = pow(y2, 2) - pow(y1, 2);
 
    float f = ((sx13) * (x12) + (sy13) * (x12) + (sx21) * (x13) + (sy21) * (x13))
            / (2 * ((y31) * (x12) - (y21) * (x13)));
    float g = ((sx13) * (y12) + (sy13) * (y12) + (sx21) * (y13) + (sy21) * (y13))
            / (2 * ((x31) * (y12) - (x21) * (y13)));
 
    float c = -pow(x1, 2) - pow(y1, 2) - 2 * g * x1 - 2 * f * y1;
    float h = -g;
    float k = -f;
    float r = sqrt(h * h + k * k - c);
 
    cout << "Centre is at (" << h << ", " << k << ")" << endl;
    cout << "Radius is " << r;
}
 
int main() {
    findCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31);
 
    return 0;
}

Dart

Translation of: C++

import 'dart:math';

void findCircle(
    double x1, double y1, double x2, double y2, double x3, double y3) {
  double x12 = x1 - x2;
  double x13 = x1 - x3;
  double y12 = y1 - y2;
  double y13 = y1 - y3;
  double y31 = y3 - y1;
  double y21 = y2 - y1;
  double x31 = x3 - x1;
  double x21 = x2 - x1;

  double sx13 = x1 * x1 - x3 * x3;
  double sy13 = y1 * y1 - y3 * y3;
  double sx21 = x2 * x2 - x1 * x1;
  double sy21 = y2 * y2 - y1 * y1;

  double f = (sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13) /
      (2 * (y31 * x12 - y21 * x13));
  double g = (sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13) /
      (2 * (x31 * y12 - x21 * y13));

  double c = -pow(x1, 2) - pow(y1, 2) - 2 * g * x1 - 2 * f * y1;
  double h = -g;
  double k = -f;
  double r = sqrt(h * h + k * k - c);

  print("Centre is at ($h, $k)");
  print("\nCheck radius as the distance between the centre and the first point: $r");
}

void main() {
  findCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31);
}

Output:

Centre is at (18.978515660148815, 16.265410797715866)

Check radius as the distance between the centre and the first point: 14.70862397833418

EasyLang

Translation of: Wren

proc circ x1 y1 x2 y2 x3 y3 . cx cy cr .
   x12 = x1 - x2
   x13 = x1 - x3
   y12 = y1 - y2
   y13 = y1 - y3
   y31 = y3 - y1
   y21 = y2 - y1
   x31 = x3 - x1
   x21 = x2 - x1
   sx13 = x1 * x1 - x3 * x3
   sy13 = y1 * y1 - y3 * y3
   sx21 = x2 * x2 - x1 * x1
   sy21 = y2 * y2 - y1 * y1
   f = (sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13) / (y31 * x12 - y21 * x13) / 2
   g = (sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13) / (x31 * y12 - x21 * y13) / 2
   c = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1
   cx = -g
   cy = -f
   cr = sqrt (cx * cx + cy * cy - c)
.
circ 22.83 2.07 14.39 30.24 33.65 17.31 cx cy cr
print "Centre: (" & cx & ", " & cy & ")  Radius: " & cr

Output:

Centre: (18.98, 16.27)  Radius: 14.71

F#

// Centre and radius of a circle passing through 3 points in a plane. Nigel Galloway: February 20th., 2024
let c (a,b) (c,d) (e,f)=(0.5*((a*a+b*b)*(f-d)+(c*c+d*d)*(b-f)+(e*e+f*f)*(d-b))/(a*(f-d)+c*(b-f)+e*(d-b)),
                         0.5*((a*a+b*b)*(e-c)+(c*c+d*d)*(a-e)+(e*e+f*f)*(c-a))/(b*(e-c)+d*(a-e)+f*(c-a)))
let d n g = let n,g=fst n-fst g,snd n-snd g in sqrt(n*n+g*g)
let circ P1 P2 P3 = let c=c P1 P2 P3 in (c,d c P1)

let centre,radius=circ (22.83, 2.07) (14.39, 30.24) (33.65, 17.31)
printfn $"Centre = %A{centre}, radius = %f{radius}"

Output:

Centre = (18.97851566, 16.2654108), radius = 14.708624

FreeBASIC

Translation of: Wren

Sub findCircle(x1 As Double, y1 As Double, x2 As Double, y2 As Double, x3 As Double, y3 As Double)
    Dim x12 As Double = x1 - x2
    Dim x13 As Double = x1 - x3
    Dim y12 As Double = y1 - y2
    Dim y13 As Double = y1 - y3
    Dim y31 As Double = y3 - y1
    Dim y21 As Double = y2 - y1
    Dim x31 As Double = x3 - x1
    Dim x21 As Double = x2 - x1
    
    Dim sx13 As Double = x1 * x1 - x3 * x3
    Dim sy13 As Double = y1 * y1 - y3 * y3
    Dim sx21 As Double = x2 * x2 - x1 * x1
    Dim sy21 As Double = y2 * y2 - y1 * y1
    
    Dim f As Double = (sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13) / (y31 * x12 - y21 * x13) / 2
    Dim g As Double = (sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13) / (x31 * y12 - x21 * y13) / 2
    
    Dim c As Double = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1
    Dim h As Double = -g
    Dim k As Double = -f
    Dim r As Double = Sqr(h * h + k * k - c)
    
    Print "Centre is at (" & h & ", " & k & ")"
    Print "Radius is"; r
    Print
    Print "Check radius as the distance between the centre and the first point:"
    Print Sqr((22.83 - h)^2 + (2.07 - k)^2)
End Sub

findCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31)

Sleep

Output:

Centre is at (18.97851566014882, 16.26541079771587)
Radius is 14.70862397833418

Check radius as the distance between the centre and the first point:
 14.70862397833418

jq

Adapted from Julia

Works with jq, the C implementation of jq

Works with gojq, the Go implementation of jq

# Emit {x,y,r} corresponding to the circle through the three points
# specified as [x,y] pairs.
def findcircle($p1; $p2; $p3):

  def assertEq($p; $q): if ($p - $q)|length < 1e-12 then . else "assertion failed: \($p) != \($q)" | error end;

  def ss($a; $b) : ($a|.*.) + ($b|.*.);
  
    $p1 as [$a,$b]
  | $p2 as [$c,$d]
  | $p3 as [$e,$f]

  | ($a - $e) as $ae
  | ($d - $b) as $db
  | ($b - $f) as $bf
  | ($e - $c) as $ec
  | ($c - $a) as $ca
  | ($f - $d) as $fd

  | ss($a; $b) as $a2b2
  | ss($c; $d) as $c2d2
  | ss($e; $f) as $e2f2

  | {x: (0.5 * ($a2b2 * $fd + $c2d2 * $bf + $e2f2 * $db) / ($a * $fd + $c * $bf + $e * $db)),
     y: (0.5 * ($a2b2 * $ec + $c2d2 * $ae + $e2f2 * $ca) / ($b * $ec + $d * $ae + $f * $ca)) }
  # any one of these should do / be nearly identical:
  | [ss(.x-$a; .y-$b), ss(.x-$c; .y-$d), ss(.x-$e; .y-$f)] as $r123
  | assertEq( $r123|max; $r123|min )
  | .r = (($r123 | add) / 3 | sqrt) ;

findcircle( [22.83, 2.07]; [14.39, 30.24]; [33.65, 17.31])
| "Centre = \([.x, .y]), radius = \(.r)"

Output:

Centre = [18.978515660148815,16.26541079771587], radius = 14.708623978334177

Julia

Translation of: Phix

Translation of: F#

function findcircle(p1, p2, p3)
    a, b = p1
    c, d = p2
    e, f = p3
    a2b2 = a * a + b * b
    ae = a - e
    db = d - b
    c2d2 = c * c + d * d
    bf = b - f
    ec = e - c
    e2f2 = e * e + f * f
    ca = c - a
    fd = f - d
    cx = 0.5 * (a2b2 * fd + c2d2 * bf + e2f2 * db) / (a * fd + c * bf + e * db)
    cy = 0.5 * (a2b2 * ec + c2d2 * ae + e2f2 * ca) / (b * ec + d * ae + f * ca)
    # any one of these should do / be nearly identical:
    r123 = [(cx-a)^2 + (cy-b)^2, (cx-c)^2 + (cy-d)^2, (cx-e)^2 + (cy-f)^2]
    @assert maximum(r123) - minimum(r123) < 1e-12
    r = sqrt(sum(r123) / length(r123))
    return (cx, cy), r
end

ctr, r = findcircle((22.83, 2.07), (14.39, 30.24), (33.65, 17.31))
println("Centre = $ctr, radius = $r")

Output:

Centre = (18.978515660148815, 16.26541079771587), radius = 14.708623978334177

Kotlin

import kotlin.math.sqrt

data class Point(val x: Double, val y: Double)

fun findCircle(p1: Point, p2: Point, p3: Point): Pair<Point, Double> {
    fun sq(x: Double) = x * x
    val centreX =
        0.5 * (
            (sq(p1.x) + sq(p1.y)) * (p3.y - p2.y) +
            (sq(p2.x) + sq(p2.y)) * (p1.y - p3.y) +
            (sq(p3.x) + sq(p3.y)) * (p2.y - p1.y)
        ) / (
            p1.x * (p3.y - p2.y) +
            p2.x * (p1.y - p3.y) +
            p3.x * (p2.y - p1.y)
        )
    val centreY =
        0.5 * (
            (sq(p1.x) + sq(p1.y)) * (p3.x - p2.x) +
            (sq(p2.x) + sq(p2.y)) * (p1.x - p3.x) +
            (sq(p3.x) + sq(p3.y)) * (p2.x - p1.x)
        ) / (
            p1.y * (p3.x - p2.x) +
            p2.y * (p1.x - p3.x) +
            p3.y * (p2.x - p1.x)
            )
    val centre = Point(centreX, centreY)
    val radius = sqrt(sq(centreX - p1.x) + sq(centreY - p1.y))
    return Pair(centre, radius)
}

fun main() {
    findCircle(Point(22.83,2.07), Point(14.39,30.24), Point(33.65,17.31))
        .let { (c, r) ->
            println("Centre = $c")
            println("Radius = $r")
        }
}

Output:

Centre = Point(x=18.978515660148815, y=16.26541079771587)
Radius = 14.70862397833418

Phix

Translation of: F#

with javascript_semantics
function circle(sequence p1, p2, p3)
    atom {a,b} = p1, {c,d} = p2, {e,f} = p3,
         a2b2 = a*a+b*b,  ae = a-e,  db = d-b,
         c2d2 = c*c+d*d,  bf = b-f,  ec = e-c,
         e2f2 = e*e+f*f,  ca = c-a,  fd = f-d,
         cx = 0.5*(a2b2*fd+c2d2*bf+e2f2*db)/(a*fd+c*bf+e*db),
         cy = 0.5*(a2b2*ec+c2d2*ae+e2f2*ca)/(b*ec+d*ae+f*ca)
    -- any one of these should do / be nearly identical:
    sequence r123 = {power(cx-a,2)+power(cy-b,2),
                     power(cx-c,2)+power(cy-d,2),
                     power(cx-e,2)+power(cy-f,2)}
    assert((max(r123)-min(r123))<1e-12)
    atom r = sqrt(average(r123))
    return {{cx,cy},r}
end function

atom {{cx,cy},r} = circle({22.83,2.07},{14.39,30.24},{33.65,17.31})
printf(1,"Centre = %.8f, %.7f, radius = %.6f\n",{cx,cy,r})

Output:

Centre = 18.97851566, 16.2654108, radius = 14.708624

Raku

Don't bother defining all the intermediate variables.

sub circle( (\𝒳ᵢ, \𝒴ᵢ), (\𝒳ⱼ, \𝒴ⱼ), (\𝒳ₖ, \𝒴ₖ) ) {
    :center(
        my $𝒞ₓ = ((𝒳ᵢ² + 𝒴ᵢ²) × (𝒴ₖ - 𝒴ⱼ) + (𝒳ⱼ² + 𝒴ⱼ²) × (𝒴ᵢ - 𝒴ₖ) + (𝒳ₖ² + 𝒴ₖ²) × (𝒴ⱼ - 𝒴ᵢ)) /
                  (𝒳ᵢ × (𝒴ₖ - 𝒴ⱼ) + 𝒳ⱼ × (𝒴ᵢ - 𝒴ₖ) + 𝒳ₖ × (𝒴ⱼ - 𝒴ᵢ)) / 2,
        my $𝒞ᵧ = ((𝒳ᵢ² + 𝒴ᵢ²) × (𝒳ₖ - 𝒳ⱼ) + (𝒳ⱼ² + 𝒴ⱼ²) × (𝒳ᵢ - 𝒳ₖ) + (𝒳ₖ² + 𝒴ₖ²) × (𝒳ⱼ - 𝒳ᵢ)) /
                  (𝒴ᵢ × (𝒳ₖ - 𝒳ⱼ) + 𝒴ⱼ × (𝒳ᵢ - 𝒳ₖ) + 𝒴ₖ × (𝒳ⱼ - 𝒳ᵢ)) / 2
    ),
    radius => (($𝒞ₓ - 𝒳ᵢ)² + ($𝒞ᵧ - 𝒴ᵢ)²).sqrt
}

say circle (22.83,2.07), (14.39,30.24), (33.65,17.31);

Output:

(center => (18.97851566 16.2654108) radius => 14.70862397833418)

You may Attempt This Online!

Swift

import Foundation
import Matrix

extension Matrix where Element: SignedNumeric {
	func minor(row: Int, column: Int) -> Element {
		var submatrix = self
		_ = submatrix.remove(rowAt: row - 1)
		_ = submatrix.remove(columnAt: column - 1)
		return submatrix.determinant as Element
	}
}

enum MatrixErrors: Error {
	case notEnoughPoints, tooManyPoints, pointsOnALine, miscError
}

func circleFrom3Points(points: (Double,Double)... ) throws -> (Double,Double,Double){
	var pointArray: [[Double]] = [[0,0,0,0]]
	for p in points {
		pointArray.append([pow(p.0, 2) + pow(p.1, 2), p.0, p.1, 1])
	}
	guard pointArray.count > 3 else { throw MatrixErrors.notEnoughPoints }
	guard pointArray.count < 5 else { throw MatrixErrors.tooManyPoints }
	var matrix = Matrix(elements:pointArray)
	var m11 = matrix.minor(row: 1, column: 1)
	guard m11 != 0 else { throw MatrixErrors.pointsOnALine }
	var m12 = matrix.minor(row: 1, column: 2)
	var m13 = matrix.minor(row: 1, column: 3)

	let x =  0.5 * m12 / m11
	let y = -0.5 * m13 / m11
	let r = (pow(x - pointArray[1][1],2) + pow(y - pointArray[1][2],2)).squareRoot()
	return (x,y,r)
}

do {
	let (x,y,r) = try circleFrom3Points(points:  (22.83,2.07), (14.39,30.24), (33.65,17.31))
	print("x:\(x), y:\(y), r: \(r)")
} catch {
	debugPrint(error)
	exit(1)
}

x:18.978515660148812, y:16.265410797715873, r: 14.708623978334185

Wren

This follows the lines of the C++ code here.

var findCircle = Fn.new { |x1, y1, x2, y2, x3, y3|
    var x12 = x1 - x2
    var x13 = x1 - x3
    var y12 = y1 - y2
    var y13 = y1 - y3
    var y31 = y3 - y1
    var y21 = y2 - y1
    var x31 = x3 - x1
    var x21 = x2 - x1

    var sx13 = x1 * x1 - x3 * x3
    var sy13 = y1 * y1 - y3 * y3
    var sx21 = x2 * x2 - x1 * x1
    var sy21 = y2 * y2 - y1 * y1

    var f = sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13
    f = f / (y31 * x12 - y21 * x13) / 2

    var g = sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13
    g = g / (x31 * y12 - x21 * y13) / 2

    var c = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1
    var h = -g
    var k = -f
    var r = (h * h + k * k - c).sqrt
    return [h, k, r]
}

var hkr = findCircle.call(22.83, 2.07, 14.39, 30.24, 33.65, 17.31)
System.print("Centre is at %([hkr[0], hkr[1]])")
System.print("Radius is %(hkr[2])")

System.print("\nCheck radius as the distance between the centre and the first point:")
System.print(((22.83 - hkr[0]).pow(2) + (2.07 - hkr[1]).pow(2)).sqrt)

Output:

Centre is at [18.978515660149, 16.265410797716]
Radius is 14.708623978334

Check radius as the distance between the centre and the first point:
14.708623978334

XPL0

Translation of: C++

proc FindCircle(X1, Y1, X2, Y2, X3, Y3);
real X1, Y1, X2, Y2, X3, Y3;
real X12, X13, Y12, Y13, Y31, Y21, X31, X21,
     SX13, SY13, SX21, SY21,
     F, G, C, H, K, R;
[
X12:= X1 - X2;
X13:= X1 - X3; 
Y12:= Y1 - Y2;
Y13:= Y1 - Y3;
Y31:= Y3 - Y1;
Y21:= Y2 - Y1;
X31:= X3 - X1;
X21:= X2 - X1;
    
SX13:= sq(X1) - sq(X3);
SY13:= sq(Y1) - sq(Y3);
SX21:= sq(X2) - sq(X1);
SY21:= sq(Y2) - sq(Y1);
 
F:= (SX13*X12 + SY13*X12 + SX21*X13 + SY21*X13) / (2.*(Y31*X12 - Y21*X13));
G:= (SX13*Y12 + SY13*Y12 + SX21*Y13 + SY21*Y13) / (2.*(X31*Y12 - X21*Y13));
 
C:= -sq(X1) - sq(Y1) - 2.*G*X1 - 2.*F*Y1;
H:= -G;
K:= -F;
R:= sqrt(H*H + K*K - C);

Text(0, "Centre is at ");  RlOut(0, H);  Text(0, ", ");  RlOut(0, K);  CrLf(0);
Text(0, "Radius is ");  RlOut(0, R);
];
 
FindCircle(22.83, 2.07, 14.39, 30.24, 33.65, 17.31)

Output:

Centre is at    18.97852,    16.26541
Radius is    14.70862