Question 1161183
.


            The solution by the tutor @math_helper is fine and correct.


            In my post,  I want to show you much shorter and more geometric solution to this problem.



<pre>
Notice that  (b-c) + (c-a) + (a-b) = 0.    (It is obvious !)


It means that x= 1 is the root to this quadratic.



But the problem states that the root is REPEATED (!)

It means that the quadratic has the minimum (or the maximum) at the point  x= 1.


In any case, it means that  the point (1,0) in the coordinate plane is the VERTEX of the quadratic function/(parabola).



It implies that  

    1 = -{{{B/(2A)}}},    (1)    (well known formula for the parabola's symmetry axis)

where "A" is the coefficient at x^2 and "B" is the coefficient at x.



In our case  A = b-c  and  B = c-a;  hence, (1) means that 


    1 = -{{{(c-a)/(2*(b-c))}}},    or

    2*(b-c) = -(c-a),

    2b - 2c = -c + a

    2b      = -c + a + 2c = a + c

     b                    = {{{(a+c)/2}}}
</pre>

I hope that after reading my solution, you will better understand, why this statement takes place.