When is perspectivity of triangles transitive?

Question

I am having a problem with Exercise 2.4.2 in the book Projective Geometry by H.S.M. Coxeter:

If two quadrangles have the same quadrangular set, then their diagonal triangles are perspective.

In my attempt I have drawn two quadrangles $ABCD$ and $NQOP$ with the same quadrangular set $(HE)(IJ)(GF)$ on the green line. The diagonal triangle for $ABCD$ is the blue triangle $MLK$, and that for $NQOP$ is the red triangle $RST$. Now I see that triangles $ABD$ and $QOP$ are perspective from the green line, tiangles $ABD$ and $MKL$ are perspective from the point $A$, and triangles $QOP$ and $TRS$ are perspective from the point $O$. How can I deduce that $MKL$ and $TRS$ are perspective from the green line (which I am guessing from the picture)?

Answer 1

$ABC$ and $OQN$ are perspective from the green line, so they are perspective from a point: $AO$, $BQ$ and $CN$ are concurrent at a point $X$.

Similarly, $ABD$ and $OQP$ are also perspective from the green line, so $AO$, $BQ$ and $DP$ are also concurrent - thus $DP$ also goes through $X$.

$BMC$ and $QRN$ are also perspective from the green line, so $BQ$, $MR$ and $CN$ are also concurrent, so $MR$ goes through $X$.

Similarly (using the perspective triangle-pairs $KBD$ with $TQP$ and $LBD$ with $SPQ$) we can prove that $TK$ and $SL$ also goes through $X$, so $MKL$ and $TRS$ are perspective from the point $X$.