@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. 

@Kitonum I have just an equation solve(abs(-2*x+5)+abs(-2*x+9)-x^2+7*x-16 = 0, x); by trying many times. Please write a procedure to solve this problem. 

@Carl Love I found one equation
solve(abs(-2*x+5)+abs(-2*x+9)-x^2+7*x-16 = 0, x);

@vv I am sorry about my edit.

@Carl Love I have just edit my question. I am sorry.

@Kitonum Thank you very much. 

@Kitonum I tried your code and copy all coordinates to LaTeX. I see that, ABCD is not a convex quadrilateral.
 

@Kitonum I find angles to draw in LaTeX.