Linearity of Power of Point 
I have 4 questions about this:
1)What is meant by $\mathbb{R}^2 \rightarrow \mathbb{R}$ in this context?
2)What is meant by $C=kA+(1−k)B$?Aren't $A,B,C$ just points in euclidean plane?How can they be related by an equation?
3)Why is it sufficient to show the condition for points $A$ and $B$?
4)What is meant by $F$ is linear?How to prove it and what is it's use?
Sorry if that's too obvious.I am not aware of the terminology used here i guess.
 A: 
1)What is meant by R2→R in this context?

$F$ is a function whose domain is $\mathbb{R}^2$ and codomain is $\mathbb{R}$. The domain is $\mathbb{R}^2$ because its inputs are points, and the codomain is $\mathbb{R}$ because its outputs are real numbers. Since what $F$ does is ignore the $\omega$ terms in $\mathbb{P}$, it is what is called a `partial function' in programming. Put another way, the domain of $\mathbb{P}$ is $\mathbb{R}^2 \times \mbox{the space of circles}^2$ (sorry, I don't know what space $\omega$ lives in), and its codomain is $\mathbb{R}$. What $F$ does is ignore some of the inputs.

2)What is meant by C=kA+(1−k)B?Aren't A,B,C just points in euclidean plane?How can they be related by an equation?

It's a statement that $C$ is on the line segment between $A$ and $B$. This kind of formula: $kx + (1-k)y$ is called a convex combination of $x$ and $y$. In the context of two points in Euclidean space, it is a way to associate the points on the line thru $x$ and $y$ with values of $k$. If $0 \leq k\leq 1$, then the point is between $x$ and $y$.

3)Why is it sufficient to show the condition for points A and B?


4)What is meant by F is linear?How to prove it and what is it's use?

The answers to 3 and 4 are connected. A function $f$ is linear if $f(a + cb) = f(a) + c f(b)$ no matter what $a$ and $b$ (and $c$) are. So the statement about sufficiency is just a bit of semantic fluff to get the proof started. Any time you want to prove a function is linear, you start with a statement similar to: "Let $a$ and $b$ be in the domain of the function".
