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[{p1=={x2-x1,y2-y1,z2-z1},p2=={x2-x3,y2-y3,z2-z3},p3=={x4-x3,y4-y3,z4-z3},p4=={x1-x4,y1-y4,z1-z4},VectorAngle[p1,{a2,b2,c2}]==VectorAngle[p2,{a2,b2,c2}],p1.({a2,b2,c2}×p2)==0,a2×x2+b2×y2+c2×z2+d2==0,VectorAngle[p2,{a3,b3,c3}]==VectorAngle[p3,{a3,b3,c3}],p2.({a3,b3,c3}×p3)==0,a3×x3+b3×y3+c3×z3+d3==0,VectorAngle[p3,{a4,b4,c4}]==VectorAngle[p4,{a4,b4,c4}],p3.({a4,b4,c4}×p4)==0,a4×x4+b4×y4+c4×z4+d4==0,VectorAngle[p1,{a1,b1,c1}]==VectorAngle[p4,{a1,b1,c1}],p1.({a1,b1,c1}×p4)==0,a1×x1+b1×y1+c1×z1+d1==0},{x1,y1,z1,x2,y2,z2,x3,y3,z3,x4,y4,z4},Reals]
