Very hard to see, but let me give it a try. But beware:how you draw the figure can make it difficult to see.

I'll make more definitions:

Let distance of L1 from M be p.

Let distance of L1 from M be q.

Let normal from P1 intersects M at A1.

Let normal from P2 intersects M at A2.

Let normal from Q1 intersects M at B1.

Let normal from Q2 intersects M at B2.

Let P1Q1 intersects M at X.

Let P1Q2 intersects M at Y.

Let P2Q1 intersects M at Z.

Let P2Q2 intersects M at T.

Clear? Are you still here?

Now (for point P2)

|ZB1|/|ZA2| = q/p (similar triangles)

|TB2|/|TA2| = q/p (similar triangles)

=> triangles: A2B1B2 & A2ZT are similar.

=> ZT is parallel to B1B2. (1)

Do the same thing for point A1:

|XB1|/|XA1| = q/p (similar triangles)

|YB2|/|YA1| = q/p (similar triangles)

=> triangles: A1B1B2 & A1XY are similar.

=> XY is parallel to B1B2. (2)

(1) and (2) => ZT is parallel to XY.

Now I will stop here. You try to see:

ZX || A1A2 and TY || A1A2

Beware:how you draw the figure can make it difficult to see.

Do you need a figure? I won't promise if I can do it. But I may try.

-O believes - deeply.