# Inequality that looks like the reverse Minkowski one (but it's not)

Let $x_1, x_2, x_2, y_1, y_2, y_3 \in \mathbb{R}^{+}$. Is it true that if $x_1^2+x_2^2<x_3^2$ and $y_1^2+y_2^2<y_3^2$ then $(x_1+y_1)^2+(x_2+y_2)^2<(x_3+y_3)^2$ ? Thanks in advance.

We will show that

$$(x_1 + y_1) ^2 + (x_2 + y_2)^2 \leq \left( \sqrt{ x_1^2 + x_2^2} + \sqrt{ y_1^2 + y_2^2} \right)^2 < (x_3 + y_3)^2$$

This is equivalent to

$$2x_1y_1 + 2 x_2 y_2 \leq 2 \sqrt{ x_1^2 + x_2^2 } \sqrt{ y_1 ^2 + y_2^2 }$$

This is equivalent to

$$2 x_1x_2y_1y_2 \leq x_1^2y_2^2 + x_2^2 y_1^2$$

This is equivalent to

$$0 \leq ( x_1 y_2 - x_2y_1)^2$$

Hence, it is true.