calculus - Cauchy-Schwarz inequality application

I am having trouble verifying the first step where the author makes use of Cauchy-Schwarz inequality.
https://gbas2010.wordpress.com/2011/10/16/inequality-53-vo-quoc-ba-can/

I am unsure how he chooses the terms from $( a + b + c )$ to square and multiple on the LHS.

Answer

He is using the two variable form of the inequality, that is

$$\begin{equation} (x^2 + y^2)(z^2 + w^2) \geq (xz + yw)^2 \end{equation}$$
Now set $x = \sqrt{a^2 + b^2}$, $y = c$, $z = \frac{a + b}{\sqrt{a^2 + b^2}}$ and $w = 1$ and we get
$$\begin{align} ((\sqrt{a^2 + b^2})^2 + c^2)((\frac{a + b}{\sqrt{a^2 + b^2}})^2 + 1 ) &\geq (\sqrt{a^2 + b^2} \frac{a + b}{\sqrt{a^2 + b^2}} + c \cdot 1)^2\\ (a^2 + b^2 + c^2)(\frac{(a + b)^2}{a^2 + b^2} + 1 ) &\geq (a + b + c)^2 \end{align}$$
Which is the first step of the solution.