Let's do it..

$a^{2}+b^{2}+c^{2}=a^{2}+ac+c^{2}=189$ .

Adding and Subtracting $ac$ in the above expression we get:

$a^{2}+2ac+c^{2}-ac=(a+c)^{2}-b^{2}=189$ .

Now, $a+c=21-b$ . Substituting the value of $a+c$ in the above equation we get:

$(21-b)^{2}-b^{2}=189$ and $b=6$ .

Now, $a^{2}+c^{2}=153$ and $ac=36$ .

$(a-c)^{2}=a^{2}+c^{2}-2ac=153-72=81$

So, $a-c=\pm{9}$ .

Taking $a-c=9$ , we get $a=12$ and $c=3$ .

Hence, the desired answer is $\lfloor{12\times 6\times 3}\rfloor=216$ .

Taking $a-c=-9$ , we get $a=3$ and $c=12$ .

Hence, the desired answer is $\lfloor{3\times 6\times 12}\rfloor=216$

So, considering both the cases our final answer is $\huge{216}$ .

Similar to Alexandra Hewitt .
$441= (a + b +c)^2 \\= a^2 + b^2 + c^2 + 2(ab + bc + c a) \\= 189 + 2(ab + bc + b^2)\\ \implies~441-189= 252\\=2b(a + b + c) \\=2b*21. ~\\ \implies~~b=6. \\ ~\therefore~a*b*c=b^3=216 . Same for floor fuction.$

ab+bc+ca=126

b(a+c)+ca=126

b(21-b)+b^2=126

so b=6 hence ac=36

so abc =216

so simple question!

a^2+ac+c^2=189 (Substituting Equation 1 to Equation 3) a^2+2ac+c^2=189+b^2(Addingboth sides by b^2: +ac in the left,+b^2 for right) (a+c)^2=189+b^2 (Simplify left side) a+c=(189+b^2)^1/2 (Derived Equation 1) b+ (189+b^2)^1/2 = 21 (Derived Equation 1 substituted to Equation 2) 189+b^2=441-42b_b^2 (Transposing b to the right side, squaring both sides) b=6 (Simplifying) b^2=ac (Equation 1) abc=b^3=216 (Equation 1)

By : Pradeep Sparkle [12 x 6 x 3] =216

Starting similar to Yash,

$a^2 + b^2 + c^2 = a^2 + ac + c^2 = 189$ .

Completing the square,

$(a+c)^2-ac = 189$ .

Substituting equation 1,

$(a+c)^2-b^2 = 189$ .

Using difference of squares,

$(a+c+b)(a+c-b) = 189$ .

Dividing by $a+b+c = 21$ ,

$a+c-b = 9$ .

Adding this to $a+b+c$ and dividing by 2 yields $a+c = 15$ , so $b = 21 - 15 = 6$ . Since we want $abc = b^2 * b = b^3$ , our answer is $\boxed{216}$ .

I can't work out how to start a new line! But here's the clearest I can format it: $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(ab+bc+ca)$ So $441=189+2(ab+bc+b^{2})$ So $252=2b(a+b+c)$ So $126=21b$ So $b+6$ So $b^{2}=36=ac$ So $abc=6x36=216$

Thanx to you now i know what a Floor and Ceiling Functions means.. Good one though..

a=12 ,b=6 ,c=3 abc=216