證明：?$(1)$?在四邊形?$ABCD$?中
∵?$E、$??$F、$??$G、$??$H$?分別是?$AD、$??$BC、$??$BD、$??$AC$?的中點(diǎn)，
∴?$FG= \frac {1}{2}CD，$??$HE =\frac {1}{2}CD，$??$FH=\frac {1}{2}AB，$??$GE=\frac {1}{2}AB$?
∵?$AB = CD$?
∴?$FG=FH= HE= EG.$?
∴四邊形?$EGFH$?是菱形
?$(2)$?解：在四邊形?$ABCD$?中，?$G，$??$F，$??$H$?分別是?$BD、$??$BC、$??$AC$?的中點(diǎn)
∴?$GF //DC，$??$HF//AB$?
∴?$∠GFB=∠DCB，$??$∠HFC=∠ABC$?
∴?$∠HFC+∠GFB =∠ABC+∠DCB =90°$?
∴?$∠GFH=90°$?
∴菱形?$EGFH$?是正方形
∵?$AB=1$?
∴?$EG=\frac {1}{2}AB=\frac {1}{2}$?
∴正方形?$EGFH$?的面積?$=(\frac {1}{2})2=\frac {1}{4}$?
