In[]:=
a4=0.2357228604996813;​​b4=-0.0468491667698241;​​c4=-0.2501467060062771;​​d4=1;​​​​a3=-0.01782102389800156;​​b3=0.010633914885299765;​​c3=-0.01601855451223666;​​d3=1;​​​​a2=-0.01070143753058265;​​b2=-0.006388724903186782;​​c2=0.0096270086182365;​​d2=1;​​​​a1=0.020725615125472996;​​b1=0.004119754753445611;​​c1=0.021977733122801962;​​d1=1;​​​​FindInstance[​​{​​({x2-x1,y2-y1,z2-z1}-{x2-x3,y2-y3,z2-z3}).{a2,b2,c2}==0,​​{x2-x1,y2-y1,z2-z1}.({a2,b2,c2}×{x2-x3,y2-y3,z2-z3})==0,​​a2×x2+b2×y2+c2×z2+d2==0,​​({x2-x3,y2-y3,z2-z3}-{x4-x3,y4-y3,z4-z3}).{a3,b3,c3}==0,​​{x2-x3,y2-y3,z2-z3}.({a3,b3,c3}×{x4-x3,y4-y3,z4-z3})==0,​​a3×x3+b3×y3+c3×z3+d3==0,​​({x4-x3,y4-y3,z4-z3}-{x1-x4,y1-y4,z1-z4}).{a4,b4,c4}==0,​​{x4-x3,y4-y3,z4-z3}.({a4,b4,c4}×{x1-x4,y1-y4,z1-z4})==0,​​a4×x4+b4×y4+c4×z4+d4==0,​​({x2-x1,y2-y1,z2-z1}-{x1-x4,y1-y4,z1-z4}).{a1,b1,c1}==0,​​{x2-x1,y2-y1,z2-z1}.({a1,b1,c1}×{x1-x4,y1-y4,z1-z4})==0,​​a1×x1+b1×y1+c1×z1+d1==0​​},​​{x1,y1,z1,x2,y2,z2,x3,y3,z3,x4,y4,z4},​​Reals​​]
Out[]=
{{x1440.,y1-6144.,z18384.,x2512.,y210752.,z2-1456.,x32176.,y315872.,z33328.,x4-243.108,y4-8320.,z43056.}}