Three spheres are tangent to a plane at the vertices of a triangle and are tangent to each other. Denote the points where spheres are tangent to the lane A, B, C. You get a triangle ABC with lengths of sides 6, 8 and 10. Denote [tex]C_A, C_B, C_C[/tex] the projections of the tangent points between spheres on the pane.
Then
[tex]CC_B=CC_A=c', \\ AC_B=AC_C=a', \\ BC_A=BC_C=b'[/tex]
and
[tex]a'+b'=6, \\ a'+c'=8, \\ b'+c'=10[/tex].
Add all these three equations: [tex]2a'+2b'+2c'=24, a'+b'+c'=12[/tex] and substract each equation from [tex]a'+b'+c'=12[/tex]:
[tex]a'+b'+c'-(a'+b')=12-6, \\ a'+b'+c'-(a'+c')=12-8, \\ a'+b'+c'-(b'+c')=12-10[/tex].
Then
[tex]c'=6, \\ b'=4, \\ a'=2
[/tex] are spheres' radii.
