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.