Gdddk

Is there a smarter way to animate a collection of possible planes passing through 2 given points P on ABFE and Q on DCGH such that we can visually prove the point R is the common point of ABCD and PQ?

documentclass[pstricks,border=12pt,12pt]standalone
usepackagepst-eucl

begindocument
% first frame
beginpspicture[showgrid=false](-1,-3)(13,14)
 pstGeonode[PosAngle=180,0,0,135,180](0,0)A(10,0)B(12,3)C(2,3)D(0,10)E
 pstTranslation[PosAngle=0,0,135]AEB,C,D[F,G,H]
 psline(E)(H)(G)(C)(B)(F)(E)(A)(B)
 psline(F)(G)
 psline[linestyle=dashed](H)(D)(C)
 psline[linestyle=dashed](A)(D)
 % Extra
 pstGeonode[PosAngle=180,-90](1,9)P(7,-2)Q
 pstTranslation[PosAngle=180,PointName=none,PointSymbol=none]EAP
 pstTranslation[PointName=none,PointSymbol=none]BFQ
 pstInterLL[PosAngle=-90]ABPP'X
 pstInterLL[PosAngle=45]CDQQ'Y
 pstTranslation[PosAngle=90]AEX,Y
 pspolygon[linestyle=none,fillstyle=solid,fillcolor=yellow,opacity=0.25](X)(X')(Y')(Y)
 pspolygon[linestyle=none,fillstyle=solid,fillcolor=cyan,opacity=0.25](A)(B)(C)(D)
 psline[linecolor=red,linestyle=dashed](X)(Y)
 psline[linecolor=red,linestyle=dashed](P)(Q)
 pstInterLL[PosAngle=0]PQXYR
endpspicture

% second frame
beginpspicture[showgrid=false](-1,-3)(13,14)
 pstGeonode[PosAngle=180,0,0,135,180](0,0)A(10,0)B(12,3)C(2,3)D(0,10)E
 pstTranslation[PosAngle=0,0,135]AEB,C,D[F,G,H]
 psline(E)(H)(G)(C)(B)(F)(E)(A)(B)
 psline(F)(G)
 psline[linestyle=dashed](H)(D)(C)
 psline[linestyle=dashed](A)(D)
 % Extra
 pstGeonode[PosAngle=180,-90](1,9)P(7,-2)Q
 pstTranslation[PosAngle=180,PointName=none,PointSymbol=none]EHP
 pstTranslation[PointName=none,PointSymbol=none]CBQ
 pstInterLL[PosAngle=-90]ABPQ'X
 pstInterLL[PosAngle=45]CDQP'Y
 pstInterLL[PosAngle=-90]EFPQ'X'
 pstInterLL[PosAngle=45]HGQP'Y'
 pspolygon[linestyle=none,fillstyle=solid,fillcolor=yellow,opacity=0.25](X)(X')(Y')(Y)
 pspolygon[linestyle=none,fillstyle=solid,fillcolor=cyan,opacity=0.25](A)(B)(C)(D)
 psline[linecolor=red,linestyle=dashed](X)(Y)
 psline[linecolor=red,linestyle=dashed](P)(Q)
 pstInterLL[PosAngle=0]PQXYR
endpspicture
enddocument

I may not do an animation and just use a plane with a nontrivial opacity such that one can see that the dashed red line hits the plane. My take is that you want to

visually prove the point R is the common point of ABCD and PQ

because this is what is written in the question. If that's not what the question is about, consider editing it to make it clearer. (Sorry, I have quit pstricks a while ago, but I am sure you will be able to redo this with pstricks.)

documentclass[tikz,border=3.14mm]standalone 
usepackagetikz-3dplot 
usetikzlibrarybackgrounds 
newcountercoord
begindocument 
tdplotsetmaincoords70200 
begintikzpicture[tdplot_main_coords,scale=3,bullet/.style=fill,circle,inner sep=1pt] 
 foreach Z in -1,1
 foreach Y in -1,1
 foreach X in -1,1 
 stepcountercoord
 path (Y*X,-1*Y,Z) coordinate (Alphcoord) -- ++ (0.2*Y*X,0,0) 
 node$Alphcoord$ ;
 fill[cyan,opacity=0.5] (A) -- (B) -- (C) -- (D) -- cycle;
 draw[dashed] (A) -- (D) -- (C) (D) -- (H); 
 draw (E) -- (F) -- (G) -- (H) -- (E) -- (A) -- (B) -- (C) -- (G) (B) -- (F);
 node[bullet,label=left:$P$] (P) at (1,0,0.5);
 node[bullet,label=above right:$R$] (R) at (1/3,0,-1);
 draw[red,dashed,thick] (P) -- (R)
 node[pos=1.5,black,bullet,label=[black]below:$Q$] (Q); 
 beginscope[on background layer]
 draw[red,dashed,thick] (R) -- (Q);
 endscope
endtikzpicture
enddocument