# Thread: Prove the sum of a² and d² is equal the sum of b² and c².

1. Let $a,\,b,\,c,\,d\in \mathbb{R}$ with condition that $a+b\sqrt{2}+c\sqrt{3}+2d\ge \sqrt{10(a^2+b^2+c^2+d^2)}$.

Prove that $a^2+d^2=b^2+c^2$.

Originally Posted by anemone
Let $a,\,b,\,c,\,d\in \mathbb{R}$ with condition that $a+b\sqrt{2}+c\sqrt{3}+2d\ge \sqrt{10(a^2+b^2+c^2+d^2)}$.

Prove that $a^2+d^2=b^2+c^2$.
Hint:

3. Originally Posted by anemone

Thanks kaliprasad for participating and the nice solution, I solved it a bit different than yours:

5. Originally Posted by anemone
Thanks kaliprasad for participating and the nice solution, I solved it a bit different than yours:

Argh...sorry kaliprasad, last night I wasn't feeling okay and therefore the solution that I posted (with typo) in a hurry didn't quite hold water.

I therefore want to give the reasoning that I've "in my mind" that would definitely work, to amend the incomplete solution I posted yesterday: