Triangle of Circular Arcs

Let $r_a$, $r_b$ and $r_c$ be the radii of three circles that are mutually and externally tangent to each other. The three circles then form a triangle of circular arcs between their tangency points as shown for the three blue circles in the picture below.

Define the circumcircle of this triangle to be the red circle, with centre $D$, passing through their tangency points. Further define the incircle of this triangle to be the green circle, with centre $E$, that is mutually and externally tangent to all the three blue circles. Let $d=\vert DE \vert$ be the distance between the centres of the circumcircle and the incircle.

Let $\mathbb{E}(d)$ be the expected value of $d$ when $r_a$, $r_b$ and $r_c$ are integers chosen uniformly such that $1\leq r_a

To solve this problem, we need to find the expected value of the distance between the centers of the circumcircle and the incircle, given certain constraints on the radii of the circles.

In order to find $\mathbb{E}(d)$, we can simulate the process of choosing random integers for the radii and compute the distance $d$ for each set of radii. By averaging these distances over a large number of simulations, we can approximate the expected value $\mathbb{E}(d)$.

Here’s a step-by-step Python code to solve this problem:

“`python
import math
import random

def calculate_distance(ra, rb, rc):
# Given the radii of the three circles, calculate the distance between their centers

# Calculate the semi-perimeter of the triangle formed by the radii
s = (ra + rb + rc) / 2

# Calculate the inradius of the triangle
inradius = math.sqrt((s – ra) * (s – rb) * (s – rc) / s)

# Calculate the distance between the centers of the circumcircle and the incircle
distance = math.sqrt(ra**2 + inradius**2) – inradius

return distance

def simulate_expected_value(num_simulations):
total_distance = 0

for _ in range(num_simulations):
# Choose random radii: ra < rb < rc ra = random.randint(1, 100) rb = random.randint(ra + 1, 100) rc = random.randint(rb + 1, 100) # Check if gcd(ra, rb, rc) = 1 if math.gcd(ra, math.gcd(rb, rc)) == 1: distance = calculate_distance(ra, rb, rc) total_distance += distance # Calculate the expected value expected_value = total_distance / num_simulations return expected_value # Set the number of simulations to perform num_simulations = 100000 # Simulate the expected value expected_value = simulate_expected_value(num_simulations) # Round the result to eight decimal places and print it print("{:.8f}".format(expected_value)) ``` Running this code will perform the simulations and output the expected value of $d$ rounded to eight decimal places. The more simulations you run, the more accurate the result will be.

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