Question

Three spheres are tangent to a plane at the vertices of a triangle and are tangent to each other. find the radii of these spheres if the sides of the triangle are 6, 8, and 10.

Answer 1

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 $C_A, C_B, C_C$ the projections of the tangent points between spheres on the pane.
Then 
$CC_B=CC_A=c', \\ AC_B=AC_C=a', \\ BC_A=BC_C=b'$ 
and 
$a'+b'=6, \\ a'+c'=8, \\ b'+c'=10$.

Add all these three equations: $2a'+2b'+2c'=24, a'+b'+c'=12$ and substract each equation from  $a'+b'+c'=12$:

$a'+b'+c'-(a'+b')=12-6, \\ a'+b'+c'-(a'+c')=12-8, \\ a'+b'+c'-(b'+c')=12-10$.
Then
$c'=6, \\ b'=4, \\ a'=2$ are spheres' radii.