If $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$, then prove that $(a^2+b^2+c^2)(d^2+b^2+c^2)=(ab+bc+cd)^2$

Question

It is given that $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$.

So how will I prove that $(a^2+b^2+c^2)(d^2+b^2+c^2)=(ab+bc+cd)^2$?

This page contains information on how to properly type mathematics on this site. — JMoravitz, Jun 17 '16 at 20:11
But I does use $ $ around your mathematical expressions and use \frac{a}{b} for a/b. Use ^ for power. Is it what you need though? — Ahmed S. Attaalla, Jun 17 '16 at 20:14
Please confirm that the current way the problem reads is the correct way it should be written. Whoever made the first edit had a fraction for the final line which would make the statement false. — JMoravitz, Jun 17 '16 at 20:17
See also: Continued proportion implies $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2$. — Martin Sleziak, Jul 08 '20 at 13:37

score 5 · Accepted Answer · answered Jun 17 '16 at 20:20

5

Let $k=\frac{c}{d}$. Then $c=kd,b=k^2d,a=k^3d$. Now simply substitute and simplify.

answered Jun 17 '16 at 20:20

1 Answers1