Consider Pappus' theorem with the unusual notation given below:
Here, two straight lines, $a$ and $b$ are given. On $a$ there are some points $AE$, $DM$, and $NB$. On $b$ there are some other points $BD$, $EN$, and $AM$. Connect the points with the red, the yellow, and the blue lines called $D$, $E$, $B$, $A$, and $M$, $N$. (The letters used to denote the lines are the common letters of the notation of the points connected.) Then, $DE$ denotes the intersection of $D$ and $E$; $BA$ denotes the intersection of $B$ and $A$; and $MN$ denotes the intersection of $M$ and $N$. According to Pappus' theorem $DE$, $BA$, and $MN$ sit on a straight line called $c$.
Now, let's use the principle of duality and let $A$, $B$, $D$, $E$, $M$, $a$, $b$, and c, and $M$ denote points. Then, translate Pappus' theorem as shown below.
The result of the translation is exactly the OP's theorem, the dual of Pappus's theore to be proved.