Python: Compute the radius and the central coordinate (x, y) of a circle which is constructed by three given points on the plane surface

Last update on April 17 2025 13:05:50 (UTC/GMT +8 hours)

Circle from Three Points

Write a program to compute the radius and the central coordinate (x, y) of a circle which is constructed from three given points on the plane surface.

Input:
x1, y1, x2, y2, x3, y3 separated by a single space.

Sample Solution:

Python Code:

# Prompt the user to input three coordinates of the circle
print("Input three coordinates of the circle:")
x1, y1, x2, y2, x3, y3 = map(float, input().split())

# Calculate distances between points
c = (x1 - x2)**2 + (y1 - y2)**2
a = (x2 - x3)**2 + (y2 - y3)**2
b = (x3 - x1)**2 + (y3 - y1)**2

# Calculate the cross-product
s = 2 * (a * b + b * c + c * a) - (a**2 + b**2 + c**2)

# Calculate the central coordinate of the circle
px = (a * (b + c - a) * x1 + b * (c + a - b) * x2 + c * (a + b - c) * x3) / s
py = (a * (b + c - a) * y1 + b * (c + a - b) * y2 + c * (a + b - c) * y3) / s

# Calculate the radius of the circle
ar = a**0.5
br = b**0.5
cr = c**0.5
r = ar * br * cr / ((ar + br + cr) * (-ar + br + cr) * (ar - br + cr) * (ar + br - cr))**0.5

# Print the radius and central coordinate of the circle
print("Radius of the said circle:")
print("{:>.3f}".format(r))
print("Central coordinate (x, y) of the circle:")
print("{:>.3f}".format(px), "{:>.3f}".format(py))

Sample Output:

Input three coordinate of the circle:
 9 3 6 8 3 6
Radius of the said circle:
3.358
Central coordinate (x, y) of the circle:
6.071 4.643

Explanation:

The above Python code takes as input three pairs of coordinates representing the points on a circle. It then calculates the radius and central coordinate of the circle based on these input coordinates using geometric formulas. Here's a step-by-step explanation:

Input Coordinates:

The user is prompted to input three pairs of coordinates (x, y) representing points on a circle.

Calculate Distances:

The code calculates the squared distances between pairs of points using the distance formula.

Calculate Cross-Product and Central Coordinate:

It computes a cross-product and uses it to calculate the central coordinates (px, py) of the circle.

Calculate Radius:

The code computes the radius of the circle using the squared distances between points.

Print Results:

The radius and central coordinates of the circle are printed with three decimal places.

Flowchart:

For more Practice: Solve these Related Problems:

Write a Python program to calculate the circumcenter and circumradius of a triangle given its three vertices.
Write a Python program to compute the intersection points of two circles given their centers and radii.
Write a Python program to determine the circle parameters (center and radius) given the endpoints of a diameter.
Write a Python program to find the circle that is tangent to two given lines and passes through a specified point.

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Python Code Editor:

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