@vv Thank you very much.
 

@vv I got "Error, invalid power". I use Maple 2018.

@Kitonum Please give Another sphere, that is mean, vertices have not the form (0,0,0), (a, 0, 0), (0, b, 0) and (0,0,c). 

@tomleslie Thank you very much. Now, I understand.

@Kitonum 
 

My computer runs longer than.
 

@vv Your method very fast. 

@vv If feel, It seems lost solutions.

@vv I do not think so. Because, the sphere tangent to "edges", not tangent to "line". The edge AB is different the line AB. 

@mmcdara Please post your code.

@Gillee Thank you very much.

@Kitonum I saw at here https://mathematica.stackexchange.com/questions/212808/find-integers-a-b-c-d-m-n-p-so-equation-has-six-distinct-solutions

@Carl Love Thank you very much.

@Carl Love Thank you very much.

@Kitonum Thank you very much.

@Kitonum I want to find some equation with 6 another solutions.