试题分析:(1)过点G作GM⊥BC于M,根据正方形的性质及同角的余角相等可证得△AHE≌△BEF,同理可证:△MFG≌△BEF,即可得到GM=BF=AE=2,再根据三角形的面积公式求解即可;
(2)过点G作GM⊥BC于M.连接HF,根据平行线的性质可得∠AHF=∠MFH,∠EHF=∠GFH,即得∠AHE=∠MFG,再结合∠A=∠GMF=90°,EH=GF可证得△AHE≌△MFG,即可得到GM=AE=2,再根据三角形的面积公式求解即可;
(3)若S
△GFC=2,则12-a=2,解得a=10.此时在△BEF中,根据勾股定理求得EF的长,在△AHE中,根据勾股定理求得AH的长,由AH>AD,即点H已经不在边AB上,故不可能有S
△GFC=2.
(1)过点G作GM⊥BC于M

在正方形EFGH中,∠HEF=90°,EH=EF,
∴∠AEH+∠BEF=90°,
∵∠AEH+∠AHE=90°,
∴∠AHE=∠BEF,
又∵∠A=∠B=90°,
∴△AHE≌△BEF.
同理可证:△MFG≌△BEF,
∴GM=BF=AE=2,
∴FC=BC-BF=10,
则S
△GFC=10;
(2)过点G作GM⊥BC于M.连接HF

∵AD∥BC,
∴∠AHF=∠MFH,
∵EH∥FG,
∴∠EHF=∠GFH,
∴∠AHE=∠MFG.
又∵∠A=∠GMF=90°,EH=GF,
∴△AHE≌△MFG.
∴GM=AE=2.
∴S
△GFC=

FC•GM=

(12-a)×2=12-a;
(3)△GFC的面积不能等于2.
∵若S
△GFC=2,则12-a=2,解得a=10.
此时,在△BEF中,EF=

=

=

,
在△AHE中,AH=

=

=

=

>12,
∴AH>AD,即点H已经不在边AB上,故不可能有S
△GFC=2.
点评:此类问题综合性强,难度较大,在中考中比较常见,一般作为压轴题,题目比较典型.