The Perimeter isn't as Easy as You Think!

Using Completing the Square to the equation $bc = a^2 -12a + 52$ , turn the equation into

$bc = (a-6)^2 + 16$

From the equation, we can know that $bc \geqslant 16$ .

Similarly, from the equation $b + c = 8$ , we can know that $bc \leqslant 16$ .

From the two equations

$bc \geqslant 16$

$bc \leqslant 16$

We can know that $bc=16$

$bc = (a-6)^2 + 16$

$(a-6)^2 = 0$

$a=6$

$\therefore a + b + c = 14$