Embedded Files
ฅ՞>ﻌ<՞ฅ Woof! 🤍❤️🧡💛💚💙💜🖤
Last edit: 29 Aug 2026
Problem: The Small Middle Circle
Problem: The Small Middle Circle
Suppose there are 3 circles all tangent to each other with circle A having radius 1 unit, circle B having radius 2 units and circle C having radius 3 units. What is the radius of the 4th circle D in the middle where it tangents to circle A, B and C?
Hold on. Let's pretend we don't know Descartes' theorem and solve it with geometry and algebra alone.
Interestingly, by connecting their centres, perfect! We got (3, 4, 5) a right angle triangle!
Plot them on xy-plane.
Finding Point A
Finding Point A
Let point A be (x, y). Thus:
A (x, y)
B (5, 0)
C (0, 0)
Knowing A is 4 units away from origin:
x² + y² = 4²
x² + y² = 16
Knowing distance AB = 3
(x - 5)²+ (y - 0)² = 3²
x² - 10x + 25 + y² = 9
x² + y² = 10x - 16
Thus,
16 = 10x - 16
x = 3.2
From x² + y² = 16:
3.2² + y² = 16
y = 2.4
Rewriting:
Point A (3.2, 2.4)
Point B (5, 0)
Point C (0, 0)
Finding Point D
Finding Point D
Let radius of circle D be r. Let point D be (x, y).
Because circle D touches circle A, whose radius is 1, the distance between their centers is r + 1. Then:
(x - 3.2)² + (y - 2.4)² = (r + 1)²
Similarly, circle D touches circle B, whose radius is 2, the distance between their centers is r + 2. Then:
(x - 5)² + (y - 0)² = (r + 2)²
(x - 5)² + y² = (r + 2)²
Also, circle D touches circle C, whose radius is 3, the distance between their centers is r + 3. Then:
(x - 0)² + (y - 0)² = (r + 3)²
x² + y² = (r + 3)²
We now have three equations and three unknowns - x, y and r.
Expand LHS of (x - 5)² + y² = (r + 2)²:
x² - 10x + 25 + y² = (r + 2)²
x² + y² = (r + 2)² + 10x - 25
Substituting in x² + y² = (r + 3)²:
(r + 3)² = (r + 2)² + 10x - 25
r² + 6r + 9 = r² + 4r + 4 + 10x - 25
2r = 10x - 30
r = 5x - 15
x = (r + 15) / 5
x = r/5 + 3
Expand LHS of (x - 3.2)² + (y - 2.4)² = (r + 1)²:
x² - 6.4x + 10.24 + y² - 4.8y + 5.76 = (r + 1)²
x² + y² = (r + 1)² + 6.4x + 4.8y - 16
Substituting in x² + y² = (r + 3)²:
(r + 3)² = (r + 1)² + 6.4x + 4.8y - 16
r² + 6r + 9 = r² + 2r + 1 + 6.4x + 4.8y - 16
4r = 6.4x + 4.8y - 24
r = 1.6x + 1.2y - 6
Substituting in x = (r + 15) / 5:
r = 1.6 * (r + 15) / 5 + 1.2y - 6
r = 0.32r + 4.8 + 1.2y - 6
0.68r = 1.2y - 1.2
34r / 60 = y - 1
y = 17r/30 + 1
Thus, point D (r/5 + 3, 17r/30 + 1).
Finding Radius r of Circle D
Finding Radius r of Circle D
Knowing distance between D and origin is r + 3:
(r/5 + 3)² + (17r/30 + 1)² = (r + 3)²
r²/25 + 6r/5 + 9 + 289r²/900 + 17r/15 + 1 = r² + 6r + 9
23r²/36 + 11r/3 - 1 = 0
23r² + 132r - 36 = 0
Solve quadratic:
r = (-132 ±√( 132² -4(23)(-36) ) ) / (2 * 23)
r = (-132 ±√20736) / 46
r = (-132 ± 144) / 46
Since r > 0:
r = (-132 + 144) / 46
r = 6 / 23
Thus, in between 3 circles of radius 1, 2, 3 units respectively, that tangent to each other, the circle that fits in the middle has a radius of 6 / 23 units.
What about a general solution?
What about a general solution?
Let:
a = radius of circle A
b = radius of circle B
c = radius of circle C
α = Angle(C,A,B)
β = Angle(A,B,C)
γ = Angle(A,C,B)
Let point C be sitting at origin (0, 0).
Finding Point A
Finding Point A
Let point A be (x, y). Thus:
A (x, y)
B (b + c, 0)
C (0, 0)
Knowing A is (a + c) away from the origin:
x² + y² = (a + c)²
Distance AB = (a + b):
(x - (b + c))² + (y - 0)² = (a + b)²
(x - (b + c))² + y² = (a + b)²
Subtract the 2 equations from one another:
(x - (b + c))² - x² = (a + b)² - (a + c)²
x² - 2x (b + c) + (b + c)² - x² = (a + b)² - (a + c)²
- 2x (b + c) + (b + c)² = (a + b)² - (a + c)²
- 2x (b + c) + (b² + 2bc + c²) = (a² + 2ab + b²) - (a² + 2ac + c²)
- 2x (b + c) + 2bc + c² = 2ab - 2ac - c²
- 2x (b + c) = 2ab - 2ac - 2bc - 2c²
x (b + c) = c² + ac + bc - ab
x (b + c) = c(c + a) + b(c - a)
x = (c(a + c) + b(c - a)) / (b + c)
From x² + y² = (a + c)²:
y² = (a + c)² - x²
Let's not expand them to keep manageable. So, Point A = (x, y) where:
x = (c(a + c) + b(c - a)) / (b + c)
y² = (a + c)² - x²
Finding Point D
Finding Point D
Let centre of circle D = (u, v) and it has a radius r.
Because circle D touches circle A, whose radius is a, the distance between their centers is r + a. Then:
(u - x)² + (v - y)² = (r + a)² ~ Equation 1
Similarly, circle D touches circle B, whose radius is b, the distance between their centers is r + b. Then:
(u - (b + c))² + (v - 0)² = (r + b)²
(u - (b + c))² + v² = (r + b)² ~ Equation 2
Also, circle D touches circle C, whose radius is c, the distance between their centers is r + c. Then:
(u - 0)² + (v - 0)² = (r + c)²
u² + v² = (r + c)² ~ Equation 3
Equation 2 - Equation 3:
(u - (b + c))² - u² = (r + b)² - (r + c)²
u² - 2u(b + c) + (b + c)² - u² = r² + 2rb + b² - r² - 2rc - c²
- 2u(b + c) + (b + c)² = 2rb + b² - 2rc - c²
- 2u(b + c) + (b + c)² = 2r(b - c) + b²- c²
(b + c)² - (b²- c²) = 2r(b - c) + 2u(b + c)
Since (b + c)² - (b²- c²) = 2bc + 2c²:
(b + c)² - (b²- c²) = 2c(b + c)
Substituting back:
2c(b + c) = 2r(b - c) + 2u(b + c)
c(b + c) = r(b - c) + u(b + c)
u = (c(b + c) + r(c - b)) / (b + c)
Equation 1 - Equation 3:
(u - x)² + (v - y)² - (u² + v²) = (r + a)² - (r + c)²
u² - 2ux + x² + v² - 2vy + y² - u² - v² = r² + 2ra + a² - r² - 2rc - c²
- 2ux + x² - 2vy + y² = 2ra + a² - 2rc - c²
x² + y² - 2ux - 2vy = 2r(a - c) + a² - c²
Substituting in x² + y² = (a + c)²:
(a + c)² - 2ux - 2vy = 2r(a - c) + a² - c²
a² + 2ac + c² - 2ux - 2vy = 2r(a - c) + a² - c²
2ac + 2c² - 2ux - 2vy = 2r(a - c)
c(a + c) - ux - vy = r(a - c)
ux + vy = c(a + c) + r(c - a)
v = (c(a + c) + r(c - a) - ux) / y
Thus, D = (u, v), where:
u = (c(b + c) + r(c - b)) / (b + c)
v = (c(a + c) + r(c - a) - ux) / y
Omelette 'em All
Omelette 'em All
This is where it tends to get ugly fast.
- Eliminate v
The goal is to clear all unknowns except r, a, b, c remain. With these known:
u² + v² = (r + c)²
x = (c(a + c) + b(c - a)) / (b + c)
y² = (a + c)² - x²
u = (c(b + c) + r(c - b)) / (b + c)
v = (c(a + c) + r(c - a) - ux) / y
Rearrange v = (c(a + c) + r(c - a) - ux) / y:
vy = (c(a + c) + r(c - a) - ux)
Square both sides:
v²y² = (c(a + c) + r(c - a) - ux)²
Substitute in y² = (a + c)² - x²:
v²((a + c)² - x²) = (c(a + c) + r(c - a) - ux)²
Substitute in v² = (r + c)² - u²:
((r + c)² - u²) * ((a + c)² - x²) = [c(a + c) + r(c - a) - ux]²
This is the key equation.
- Make Substitution Less Ugly
Let:
D = (b + c)
X = c(a + c) + b(c - a)
U = c(b + c) + r(c - b)
So, x = X / D, u = U / D.
- Substitute x = X / D into ((a + c)² - x²) of the Key Equation
((a + c)² - x²) = ((a + c)² - (X / D)²)
((a + c)² - x²) = ((a + c)² - X² / D²)
((a + c)² - x²) = ((D²(a + c)² - X²) / D²)
- Substitute u = U / D into ((r + c)² - u²) of the Key Equation
((r + c)² - u²) = ((r + c)² - (U / D)²)
((r + c)² - u²) = ((r + c)² - U² / D²)
((r + c)² - u²) = ((D²(r + c)² - U²) / D²)
- Substitute u and x into the RHS of the Key Equation
The RHS of the key equation being:
[c(a + c) + r(c - a) - ux]² = [c(a + c) + r(c - a) - UX / D²]²
- Put everything into the Key Equation
We have:
((a + c)² - x²) = ((D²(a + c)² - X²) / D²)
((r + c)² - u²) = ((D²(r + c)² - U²) / D²)
[c(a + c) + r(c - a) - ux]² = [c(a + c) + r(c - a) - UX / D²]²
((r + c)² - u²) * ((a + c)² - x²) = [c(a + c) + r(c - a) - ux]²:
((D²(r + c)² - U²) / D²) * ((D²(a + c)² - X²) / D²) = [c(a + c) + r(c - a) - UX / D²]²
( (D²(r + c)² - U²) * (D²(a + c)² - X²) ) / D⁴ = [c(a + c) + r(c - a) - UX / D²]²
(D²(r + c)² - U²) * (D²(a + c)² - X²) = D⁴ [c(a + c) + r(c - a) - UX / D²]²
(D²(r + c)² - U²) * (D²(a + c)² - X²) = [D²(c(a + c) + r(c - a)) - UX]²
Since:
D = (b + c)
X = c(a + c) + b(c - a)
U = c(b + c) + r(c - b)
Let's break it into smaller pieces.
For the LHS:
D²(r + c)² - U² = (b + c)²(r + c)² - (c(b + c) + r(c - b))²
D²(r + c)² - U² = [(b² + 2bc + c²)(r² + 2rc + c²)] - [c²(b² + 2bc + c²) + 2rc(b + c)(c - b) + r²(c² - 2bc + b²)]
D²(r + c)² - U² = [b²r² + 2b²rc + b²c² + 2bcr² + 4bc²r + 2bc³ + c²r² + 2rc³ + c⁴ ] - [b²c² + 2bc³+ c⁴ + 2rc(c² - b²) + r²c² - 2bcr² + b²r²]
D²(r + c)² - U² = [2b²rc + 2bcr² + 4bc²r + 2rc³] - [2rc(c² - b²) - 2bcr²]
D²(r + c)² - U² = [2b²rc + 2bcr² + 4bc²r + 2rc³] - [2rc³ - 2rcb² - 2bcr²]
D²(r + c)² - U² = 4b²rc + 4bcr² + 4bc²r
D²(r + c)² - U² = 4bcr (b + c + r)
D²(a + c)² - X² = (b + c)²(a + c)² - (c(a + c) + b(c - a))²
D²(a + c)² - X² = [(b² + 2bc + c²)(a² + 2ac + c²)] - [c²(a + c)² + 2bc(a + c)(c - a) + b²(c - a)²]
D²(a + c)² - X² = [a²b² + 2ab²c + b²c² + 2a²bc + 4abc² + 2bc³ + a²c² + 2ac³ + c⁴] - [a²c² + 2ac³ + c⁴ + 2bc(c² - a²) + b²c² - 2acb² + a²b²]
D²(a + c)² - X² = [2ab²c + 2a²bc + 4abc² + 2bc³] - [2bc³ - 2a²bc) - 2acb²]
D²(a + c)² - X² = 4ab²c + 4a²bc + 4abc²
D²(a + c)² - X² = 4abc (a + b + c)
Expand 4bcr (b + c + r) * 4abc (a + b + c):
= 16a²b³c²r + 16ab⁴c²r + 32ab³c³r + 16a²b²c³r + 16ab²c⁴r + 16a²b²c²r² + 16ab³c²r² + 16ab²c³r²
= 16ab²c²r(ab + b² + 2bc + ac + c² + ar + br + cr)
For the RHS:
D²(c(a + c) + r(c - a)) = (b + c)²(c(a + c) + r(c - a))
D²(c(a + c) + r(c - a)) = (b² + 2bc + c²)(ac + c² + rc - ar)
D²(c(a + c) + r(c - a)) = b²ac + b²c² + b²rc - b²ar + 2abc² + 2bc³ + 2bc²r - 2bcar + ac³ + c⁴ + c³r - c²ar
UX = [c(a + c) + b(c - a)] * [c(b + c) + r(c - b)]
UX = [ac + c² + bc - ab] * [bc + c² + rc - br]
UX = abc² + ac³ + ac²r - acbr + bc³ +c⁴ + c³r - c²br + b²c² + bc³ + bc²r - b²cr - ab²c - abc² - abcr + ab²r
UX = ac³ + ac²r + c⁴ + c³r + b²c² - b²cr - ab²c + ab²r - 2abcr + 2 bc³
D²(c(a + c) + r(c - a)) - UX = [b²ac + b²c² + b²rc - b²ar + 2abc² + 2bc³ + 2bc²r - 2bcar + ac³ + c⁴ + c³r - c²ar] - [ac³ + ac²r + c⁴ + c³r + b²c² - b²cr - ab²c + ab²r - 2abcr + 2 bc³]
D²(c(a + c) + r(c - a)) - UX = [b²ac + b²rc - b²ar + 2abc² + 2bc²r - c²ar] - [ac²r - b²cr - ab²c + ab²r]
D²(c(a + c) + r(c - a)) - UX = 2b²ac + 2b²rc - 2b²ar + 2abc² + 2bc²r - 2c²ar
D²(c(a + c) + r(c - a)) - UX = 2(b²ac + b²rc - b²ar + abc² + bc²r - c²ar)
D²(c(a + c) + r(c - a)) - UX = 2(abc² + ab²c + b²rc - b²ar + bc²r - c²ar)
D²(c(a + c) + r(c - a)) - UX = 2(abc(b+c) + r(b²c - b²a + bc² - c²a))
D²(c(a + c) + r(c - a)) - UX = 2[abc(b+c) + r(b²(c - a) + c²(b - a))]
4a²b⁴c² + 8a²b³c³ + 4a²b²c⁴ - 8a²b⁴cr - 8a²b³c²r - 8a²b²c³r + 4a²b⁴r² + 8a²b³cr² + 4a²b²c²r² + 8ab⁴c²r + 8ab³c³r + 8ab²c⁴r - 8ab⁴cr² - 16ab³c²r² - 8ab²c³r² + 4b⁴c²r² + 8b³c³r² + 4b²c⁴r²
The Big Circle E
The Big Circle E
What about the radius of the 5th circle E, larger than all of them, where it also tangents to circle A, B and C?
Page updated
Google Sites
Report abuse