极难的数学题The hexagon ABCDEF is convex and opposite sides are parallel.Show that the triangles ACE and BDF have equal area.（用中文回答）
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极难的数学题The hexagon ABCDEF is convex and opposite sides are parallel.Show that the triangles ACE and BDF have equal area.（用中文回答）
极难的数学题
The hexagon ABCDEF is convex and opposite sides are parallel.Show that the triangles ACE and BDF have equal area.
（用中文回答）
极难的数学题The hexagon ABCDEF is convex and opposite sides are parallel.Show that the triangles ACE and BDF have equal area.（用中文回答）
在凸六边形ABCDEF内作平行四边形ABCP,CDEQ,EFAR,因为六边形三组对边平行,所以,A,P,R在一直线上,C,Q,P在一直线上,E,R,Q在一直线上.
S(ABCDEF)=S(ABCP)+S(CDEQ)+S(EFAR)+S(PQR),
S(ACE)=S(ABCP)/2+S(CDEQ)/2+S(EFAR)/2+S(PQR).
又在六边形ABCDEF内作平行四边形FABM,BCDN,DEFL,因为六边形三组对边平行,所以,F,L,M在一直线上,B,M,N在一直线上,D,N,L在一直线上.
S(ABCDEF)=S(FABM)+S(BCDN)+S(DEFL)+S(LMN),
S(BDF)=S(FABM)/2+S(BCDN)/2+S(DEFL)/2+S(LMN).
易知PQ=LM=|AB-DE|,QR=MN=|AF-CD|,RP=NL=|BC-EF|,
所以,S(PQR)=S(LMN).
S(ABCP)+S(CDEQ)+S(EFAR)=S(FABM)+S(BCDN)+S(DEFL),
S(ACE)=S(BDF).