If I'm reading this right, the red part is what he's proving with the blue and green lines, so it's not supposed to make perfect sense until those other parts are established.
The blue part is how you find the midpoint between two real numbers. If x and y are real numbers, then (x+y)/2 is the midpoint between x & y.
The green part comes from an inequality that is used in analysis frequently. The inequality states that for nonnegative numbers x and y:
The left side of the inequality is known as the "Geometric mean" and the right hand side is known as the "Arithmetic mean."
Note: There is a more general statement of the above inequality. If you're curious you can find it here Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia
Does that clear things up? Let me know. Good luck!