Lesson Three circles touching externally

Three circles touching externally

Problem 1

When two circles touch externally, the distance between their center is the sum of their radii.

It gives you the system of three equations

a + b = 3,     (1)
a + c = 4,     (2)
b + c = 3.     (3)

where a, b and c are the radii of the three circles with the centers A, B and C respectively.


To solve the system, first add all the three equations. You will get

2a + 2b + 2c = 3 + 4 + 3 = 10,

which implies 

a + b + c = 5.   (4)


Now subtract equation (3) from equation (4) (both sides). You will get a = 5-3 = 2.

    Subtract equation (2) from equation (4) (both sides). You will get b = 5-4 = 1.

Finally, subtract equation (1) from equation (4) (both sides). You will get a = 5-3 = 2.

Answer.  The radii are: 2 cm for the circle with the center at A;
         1 cm for the circle with the center at B;
         2 cm for the circle with the center at C.

Problem 2

Let the radii of these circles be "a", "b" and "c", in increasing order.

Then you have these equations

a + b = 4,   (1)
a + c = 6,   (2)
b + c = 8.   (3)

Add the three equations, (1),(2) and (3), You will get

2a + 2b + 2c = 4 + 6 + 8 = 18,

which  implies

a + b + c = 9.   (4)

Now subtract eq(1) from eq(4). You will get

c = 9-4 = 5.


Next, subtract eq(2) from eq(4). You will get

b = 9-6 = 3.


Finally, subtract eq(3) from eq(4). You will get

a = 9-8 = 1.


Thus you found the radii. They are  a= 1,  b= 3  and  c= 5.


The biggest circle has the radius of 5 cm.