Question

If b is the mean proportional between a and c such that a:b=b:c
prove that:
abc(a+b+c)^3 = (ab+bc+ac)^3

Answer 1

hmm...

Answer 2

I got it.

Answer 3

\(a:b = b:c\)

\(b^2 = ac\)

\(b = \sqrt{ac} \)

\(abc(a + b + c )^3 =\)

\(=ac\sqrt{ac}(a + b + c)^3\)

\(= \sqrt{ac}\sqrt{ac}\sqrt{ac}(a + b + c)^3\)

\(= \sqrt{ac}\sqrt{ac}\sqrt{ac}(a + b + c)(a + b + c)(a + b + c)\)

\(= (a\sqrt{ac} + b\sqrt{ac} + c\sqrt{ac})^3\)

\(= (ab + b^2 + bc)^3\)

\(= (ab + ac + bc)^3\)

\(= (ab + bc + ac )^3\)

Answer 4

@vinzkie2109
Do you understand it?

Answer 5

thanks a lot!