CPE = 1/3 CPB = 1/3 CPA=1/4 CAE=1/8 ABC

BND=1/2 BNA=1/6 BNC=1/7 BCD=1/14ABC

AMF=1/4 AMC=1/8 ABM= 1/9 ABF=1/36 ABC

AMND=ABF – BND – AMF

=1/4 ABC = 1/14 ABC = 1/36 ABC= 7/42 ABC

BEPD= BCD = CPE

= ½ ABC – 1/8 ABC = 3/8 ABC

MNP = ABC – AEC – BEPD – AMND

= ABC – 1/3 ABC – 3/8 ABC – 7/42 ABC

= 1/8 ABC

khó hiểu wá

Ta có: CE+EB=CB

=>\(EB=BC-\frac23\times BC=\frac13\times BC\)

=>CE=2xEB

=>\(EB=\frac12\times EC\)

=>\(S_{AEB}=\frac12\times S_{AEC};S_{OEB}=\frac12\times S_{OEC}\)

=>\(S_{AEB}-S_{OEB}=\frac12\times\left(S_{AEC}-S_{OEC}\right)\)

=>\(S_{AOB}=\frac12\times S_{AOC}\) (1)

DA+DB=AB

=>\(DB=AB-\frac13\times AB=\frac23\times AB\)

=>\(DB=2\times DA\)

=>\(S_{CDB}=2\times S_{CDA};S_{ODB}=2\times S_{ODA}\)

=>\(S_{CDB}-S_{ODB}=2\times\left(S_{CDA}-S_{ODA}\right)\)

=>\(S_{COB}=2\times S_{AOC}\) (2)

Từ (1),(2) suy ra \(\frac{S_{BOA}}{S_{BOC}}=\frac12:2=\frac14\)

=>\(S_{BOA}=\frac14\times S_{BOC}\)

Vì \(EB=\frac13\times BC\)

nên \(S_{OEB}=\frac13\times S_{BOC}\)

=>\(\frac{S_{EBO}}{S_{BOA}}=\frac13:\frac14=\frac43\)

=>\(\frac{OE}{OA}=\frac43\)

=>\(OE=\frac43\times OA\)

Ta có: OA+OE=AE

=>\(AE=OA+\frac43\times OA=\frac73\times OA\)

=>\(AO=\frac37\times AE\)

=>\(S_{ADO}=\frac37\times S_{ADE}\)

Vì \(AD=\frac13\times AB\) nên \(S_{ADE}=\frac13\times S_{AEB}\)

=>\(S_{ADO}=\frac37\times\frac13\times S_{AEB}=\frac17\times S_{AEB}\)

Vì BE=1/3BC

nên \(S_{AEB}=\frac13\times S_{ABC}\)

=>\(S_{ADO}=\frac17\times\frac13\times S_{ABC}=\frac{1}{21}\times S_{ABC}\)

Vì \(CE=\frac23\times CB\)

nên \(S_{CEO}=\frac23\times S_{COB}\)

Ta có: \(S_{AOB}+S_{AOC}+S_{BOC}=S_{ABC}\)

=>\(S_{ABC}=S_{AOC}+2\times S_{AOC}+\frac12\times S_{AOC}=\frac72\times S_{AOC}\)

=>\(\frac{S_{COB}}{S_{ABC}}=2:\frac72=\frac47\)

=>\(S_{BOC}=\frac47\times S_{ABC}\)

=>\(S_{CEO}=\frac23\times\frac47\times S_{ABC}=\frac{8}{21}\times S_{ABC}\)

=>\(\frac{S_{ADO}}{S_{CEO}}=\frac{1}{21}:\frac{8}{21}=\frac18\)