CG=GH=HD=DA; CG+GH+HD+DA=AC=20, so CG=GH=HD=DA=5.
EF=GD=10=GH+HD.
GĈF=AĈB=EF̂B, so sinAĈB=sinEF̂B=EB/EF=EB/10. Apply Cosine Rule:
In ΔGFC, GF2=GC2+CF2-2GC.CFcosGĈF, GF2=25+CF2-2GC.CFcosGĈF.
cosGĈF=cosEF̂B=FB/10=CF/10 (FB=CF), so 2GC.CFcosGĈF=10CF(CF/10)=CF2.
GF2=25+CF2-CF2=25, and GF=5. Since GF=DE, DE=5 and the perimeter of DEFG is 2(GD+DE)=2(10+5)=30.