I post here because I don't understand a proof in projective geometry. Actually, this is the proposition :
Let $X$ be a projective plane, $D$ and $D'$ two distinct projective lines, and $f : D \rightarrow D'$ be an homography. The set $\{ Mf(N) \cap Nf(M) \; | \; (M,N) \in D \times D \}$ is a projective line.
So, this is the proof, and where I'm stuck :
Proof : Let $A, B, C$ three distincts points in $D$, and $I = Af(B) \cap Bf(A)$, $J = Af(C) \cap Cf(A)$. Let denote by $\delta$ the line $IJ$. Let $h_1$ the central projection of center $f(A)$ from $D$ to $\delta$ and $h_2$ the central projection of center $A$ from $\delta$ to $D'$. Let $h = h_2 \circ h_1$. $h$ is a composition of homography, and then is a homography. Moreover, $h = f$ over $\{A, B, C\}$ which is a projective frame, so $h=f$.
And this is the point where I'm stuck. Cause after that, there is written : "As $A$ is arbitrary chosen on $D$, for all $(M,N) \in D \times D$, the intersection point of $Mf(N)$ and $Nf(M)$ is on $\delta$.
But I don't see why, actually to define $\delta$, we have fixed $A$ on $D$. And anyway, I don't see the connection, how we can dedude this from $f=h$.
Thank you very much for the help !