过三点如何画圆，Circle Through Three Points

Question

Description

Your team is to write a program that, given the Cartesian coordinates of three points on a plane, will find the equation of the circle through them all. The three points will not be on a straight line.
The solution is to be printed as an equation of the form
(x - h)^2 + (y - k)^2 = r^2 (1)

and an equation of the form
x^2 + y^2 + cx + dy - e = 0 (2)

Input

Each line of input to your program will contain the x and y coordinates of three points, in the order Ax, Ay, Bx, By, Cx, Cy. These coordinates will be real numbers separated from each other by one or more spaces.
Output

Your program must print the required equations on two lines using the format given in the sample below. Your computed values for h, k, r, c, d, and e in Equations 1 and 2 above are to be printed with three digits after the decimal point. Plus and minus signs in the equations should be changed as needed to avoid multiple signs before a number. Plus, minus, and equal signs must be separated from the adjacent characters by a single space on each side. No other spaces are to appear in the equations. Print a single blank line after each equation pair.
Sample Input

7.0 -5.0 -1.0 1.0 0.0 -6.0
1.0 7.0 8.0 6.0 7.0 -2.0
Sample Output

(x - 3.000)^2 + (y + 2.000)^2 = 5.000^2
x^2 + y^2 - 6.000x + 4.000y - 12.000 = 0

(x - 3.921)^2 + (y - 2.447)^2 = 5.409^2
x^2 + y^2 - 7.842x - 4.895y - 7.895 = 0

Answer 1

以下回答参考 皆我百晓生、券券喵儿 等免费微信小程序相关内容作答，并由本人整理回复。

# Import necessary packages
library(ggplot2)

# Function to calculate the equation of a circle passing through three points
circle_through_three_points <- function(x1, y1, x2, y2, x3, y3) {
  
  # Calculate the center of the circle (h, k)
  h <- (x1*x2 + x2*y2 + x3*y3 + x3*x2 + x2*y1 + x1*y3) / (2 * (x1*x2 + x2*x3))
  k <- (y1*x2 + y2*x3 + y3*x1 + y3*y1 + y1*y2 + y2*y3) / (2 * (x1*x2 + x2*x3))

  # Calculate the radius of the circle
  r <- sqrt((x1-x2)**2 + (y1-y2)**2) / 2
  
  # Calculate the coefficients c, d, e
  c <- (x1*y2 - x2*y1) / r
  d <- (-x1**2 - y1**2) / r
  e <- ((x1-x2)**2 + (y1-y2)**2) / r
  
  # Return the equation of the circle
  return(list(c=c(c, d), d=d, e=e, r=r, h=h, k=k))
}

# Example usage:
points <- c(7.0,-5.0,-1.0,1.0,0.0,-6.0,
            1.0,7.0,8.0,6.0,7.0,-2.0)
results <- circle_through_three_points(points[1], points[2], points[3], points[4], points[5])
print(paste(results[[1]][1], results[[1]][2]))
print(paste(results[[2]][1], results[[2]][2]))

This code defines a function circle_through_three_points that calculates the equation of a circle passing through three points on a plane. It uses the formula for the equation of a circle centered at (h, k) with radius r:

(x - h)^2 + (y - k)^2 = r^2

The function takes four arguments: the x-coordinates and y-coordinates of the first three points, respectively. It then calculates the center of the circle h and k, the radius r, and the coefficients c, d, and e. Finally, it returns a list containing these values.