Three Tangent Circles  Problem With Solution
A problem, on three tangent circles, is presented along with detailed solution.
Problem : In the figure below, three circles are tangent to each other and to line L. The radius of circle A is equal to 10 cm and the radius of circle B is equal to 8 cm. Find the radius of circle C.
Solution to Problem :

Let a, b and c be the radii of the three circles. We first draw the lines AA', BB' and CC' perpendicular to line L. B'C, CA' and BA" are parallel to line L.

Let x = B'C, y=A'C and z = BA". Pythagora's theorem applied to the right triangle BCB' gives
x^{ 2} + BB'^{ 2} = BC^{ 2}

Note that BB' = b  c and BC = b + c. The above equation may be written as follows
x^{ 2} + (b  c)^{ 2} = (b + c)^{ 2}

Expand the above, group like terms and solve for x
x^{ 2} = 4 b c
x = 2 Square Root( b c )

Use Pythagora's theorem to triangle AA'C to obtain
y^{ 2} + (a  c)^{ 2} = (a + c)^{ 2}

Expand, group like terms and solve for y
y^{ 2} = 4 a c
y = 2 Square Root( a c )

Use Pythagora's theorem to triangle AA"B to obtain
z^{ 2} + (a  b)^{ 2} = (a + b)^{ 2}

Expand, group and solve for z
z^{ 2} = 4 a b
z = 2 Square Root( a b )

We now use the fact that z = x + y to write
Square Root( a b ) = Square Root( b c )+ Square Root( a c )

Solve the above for c
Square Root( a b ) = Square Root( c )Square Root ( b )+ Square Root( c )Square Root( a )
Square Root( c ) = Square Root( a b ) / [ Square Root ( b )+ Square Root( a ) ]
c = ( a b ) / [ Square Root ( b )+ Square Root( a ) ]^{ 2}

Substitute a and b by their values to obtain c
c = 10 * 8 / [ Square Root ( 8 )+ Square Root( 10 ) ]^{ 2}
c = 2.2 cm (rounded to 1 decimal place)
More references to geometry problems.
Geometry Tutorials, Problems and Interactive Applets.

