4.02 Proof by deduction

Sari wants to prove that (a+b)^2 \geq 2ab for all integers a and b.

Her proof is shown below:

What should she write in the ⬚ to complete her proof?

Prove that (x+y)^2-9(x-y)^2=4(2x-y)(2y-x).

Use the method of completing the square to prove that x=\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} for all quadratics of the form ax^2+bx+c=0.

If N is an even integer, prove that \dfrac{N^2}{2} is an even integer.

Prove that angle sum of a straight line is 180 \degree.

Prove that the sum of two consecutive numbers is odd.

Prove that (a+2)^2 - (a-2)^2 is divisible by 8 for any positive whole number a.

Prove that the sum of two consecutive odd numbers is an even number.

Prove that the distance between two points (a, b) and (c, d) is given by \sqrt{(c -a)^2 + (d -b)^2}.

Prove that if we subtract 1 from a positive odd square number, the answer is always divisible by 8.

Prove that 0.\dot{1}\dot{8} is a rational number.