Question

If `a/b = b/c = c/d`,then how do I prove that `(a-d)^2 = (b-c)^2 +(c-a)^2 + (b-d)^2?`

Answer 1

See explanation

Explanation:

We want to show that:

`(a-d)^2=(b-c)^2+(c-a)^2+(b-d)^2`

Expanding the expression on the right:

`b^2-2bc+c^2+c^2-2ac+a^2+b^2-2bd+d^2`

Rearranging and combining terms:

`a^2+2b^2+2c^2-2ac-2bc-2bd+d^2`

Factoring out the 2:

`a^2+2(b^2+c^2-ac-bc-bd)+d^2`

Now let’s use the fact that `a/b=b/c`

Multiply everything by `bc`, so:

`color(red)(ac=b^2)`

Same thing for `b/c=c/d` but multiply by `cd`, so:

`color(blue)(bd=c^2)`

Same thing for `a/b=c/d` but multiply by `bd`, so:

`color(green)(ad=bc)`

Find and replace these terms in our expression:

`a^2+2(color(red)(b^2)+color(blue)(c^2)-ac-color(green)(bc)-bd)+d^2`

`a^2+2(color(red)(ac)+color(blue)(bd)-ac-color(green)(ad)-bd)+d^2`

Combine like terms inside the parentheses:

`a^2+2(-ad)+d^2=a^2-2ad+d^2`

Factor:

`(a-d)^2`

Therefore:

`(a-d)^2=(b-c)^2+(c-a)^2+(b-d)^2`