Combinatorial proof of Catalan's identity Consider the problem of tiling a board of length $n$ with squares of size $1×1$ and dominoes of size $1×2$, Let's denote $f_n$ to be the number of ways to tile this so-called ($n$)-board.Then $f_n=F_{n+1}$, where $F_n$ is the nth Fibonacci number.
Now the question is as follows:

Is there any combinatorial way to prove Catalan's identity which states:
  $$F_{n}^2-F_{n-r}F_{n+r}=(-1)^{n-r}F_r^2$$


It's easy to prove this identity using induction or using a more generalized form of that which is indeed Vajda's identity.
Cassini's identity (a special case of Catalan's identity ) has been proved using combinatorial interpretation in a few books, but I've never seen any combinatorial proof of this one.
 A: I always believed that this puzzle combined with Pick's theorem is the most visually appealing proof of the identity.  You can also think of it as a bijective proof, if you allow "geometric bijections", cf. this answer.  
A: The method used to prove Example 2.7 here (which also appears in the book by Benjamin and Quinn) easily extends to give a combinatorial proof of Catalan's identity.
A: This would not be a combinatorial proof but you may take it as an alternative method. I'm referring to the method of condensation for determinants as illustrated in this joint paper. See pages 10-11.
A: I don’t have my copy in front of me, but I am almost positive that a combinatorial proof of this identity appears in the first chapter of Proofs that Really Count: The Art of Combinatorial Proof.
A: You may simply interpret both $F_n^2$ and $F_{n-r}F_{n+r}$ as the numbers of domino tylings of $2\times (2n-2)$ rectangle with additional restrictions --- vertical cuts on place $n$ (restriction 1) or $n-r$ (restriction 2), and subtract: "tylings with restriction 1" minus "tylings with restriction 2"="tylings with restriction 1 but not 2" minus "tylings with restrictions 2 but not 1". The illustration:

