In[]:=
EQ={(a^3)b-((a-2c)^3)(b-2d)==12*145560,​​​​ab^3-(a-2c)(b-2d)^3==12*125560,​​​​(2((a-c)(b-d))^2)/((a-c)/d+(b-d)/c)==34567,​​​​ab-(a-2c)(b-2d)==1256}//Simplify
Out[]=

3
a
b1746720+
3
(a-2c)
(b-2d),a
3
b
1506720+(a-2c)
3
(b-2d)
,
2
2
(a-c)
c
2
(b-d)
d
ac-
2
c
+(b-d)d
34567,bc+ad628+2cd
In[]:=
NSolve[EQ,{a,b,c,d}]
Out[]=
{{a-44.566,b0.618279,c-41.0146,d17.4401},{a44.566,b-0.618279,c41.0146,d-17.4401},{a37.4631,b34.2618,c41.0146,d17.4401},{a-37.4631,b-34.2618,c-41.0146,d-17.4401},{a18.6362-7.6551,b-16.9098+123.203,c18.6362,d0.+123.203},{a-18.6362-7.6551,b16.9098+123.203,c-18.6362,d0.+123.203},{a18.6362+7.6551,b-16.9098-123.203,c18.6362,d0.-123.203},{a-18.6362+7.6551,b16.9098-123.203,c-18.6362,d0.-123.203},{a-36.8794,b-34.8238,c-18.7736,d-38.5878},{a36.8794,b34.8238,c18.7736,d38.5878},{a-0.667749,b42.3518,c-18.7736,d38.5878},{a0.667749,b-42.3518,c18.7736,d-38.5878},{a-18.2161-123.831,b17.3073+7.61743,c0.-123.831,d17.3073},{a18.2161-123.831,b-17.3073+7.61743,c0.-123.831,d-17.3073},{a18.2161+123.831,b-17.3073-7.61743,c0.+123.831,d-17.3073},{a-18.2161+123.831,b17.3073-7.61743,c0.+123.831,d17.3073},{a37.2628,b34.3231,c0.819865,d16.8391},{a-35.623,b0.644936,c0.819865,d-16.8391},{a35.623,b-0.644936,c-0.819865,d16.8391},{a-37.2628,b-34.3231,c-0.819865,d-16.8391},{a37.2628,b34.3231,c0.819865,d16.8391},{a0.69645,b-32.8505,c-18.1303,d0.87696},{a-0.69645,b32.8505,c18.1303,d-0.87696},{a36.9571,b34.6044,c18.1303,d0.87696},{a-36.9571,b-34.6044,c-18.1303,d-0.87696},{a37.1009,b34.373,c29.146,d17.6411},{a21.1911,b-0.909271,c29.146,d-17.6411},{a-21.1911,b0.909271,c-29.146,d17.6411},{a-21.1911,b0.909271,c-29.146,d17.6411},{a-37.1009,b-34.373,c-29.146,d-17.6411},{a37.0102,b34.4558,c19.0008,d26.9204},{a37.0102,b34.4558,c19.0008,d26.9204},{a-0.991362,b19.385,c-19.0008,d26.9204},{a-37.0102,b-34.4558,c-19.0008,d-26.9204},{a0.991362,b-19.385,c19.0008,d-26.9204}}