This is a very simple qn. on the theory of quadratic eqns. Here is the solution

 

 

 

let   y=(x^2-bc)/(2x-b-c);

 

simplifying , we get the following quadratic eqn of x -

 

 

x^2-2xy+(b+c)y-bc=0

 

now, x, being real , the discriminant of the eqn should be>=0 , which gives

 

y^2-(b+c)y + bc>=0 .

which gives

(y-b)(y-c)>=0 .

hence y can't lie between b&c . (proved)