ABCD is a parallelogram, P is any point on AC. Through P, MN is drawn parallel to BA cutting BC in M and AD in N. SR is drawn parallel to BC cutting BA in S and CD in R. Show that [ASN]+[AMR]=[ABD] (where [ . ] denotes the area of the rectilinear figure).My attempt : used base division method to find the area of ASN but I get extra variables which is tough...

Question

ABCD  is a parallelogram, P is any point on AC. Through P, MN is drawn parallel to BA cutting BC in M and AD in N. SR is drawn parallel to BC cutting BA in S and CD in R. Show that [ASN]+[AMR]=[ABD] (where   [  .  ] denotes the area of the rectilinear figure).My attempt : used base division method to find the area of ASN but I get extra variables which is tough...

Maggie Bowman · Accepted Answer

[  .  ] represents areaDraw   D  P and   B  P and using the properties of parallelogram (diagonal bisects the area)  [  P  M  R  ]  =  [  C  M  R  ]  [  A  S  N  ]  =  [  P  S  N  ]  [  D  P  R  ]  =  [  A  P  R  ] (same base, equal height) and   [  D  P  R  ]  =  [  N  P  D  ]  [  B  P  M  ]  =  [  A  P  M  ] (same base, equal height) and   [  B  P  M  ]  =  [  S  P  B  ]  [  P  M  R  ]  +  [  A  S  N  ]  +  [  A  P  R  ]  =  [  A  P  M  ]  =            [      A      B      C      D      ]        2    =  [  A  B  D  ]

Keenan Santos · Accepted Answer

Note   [  A  S  N  ]  =      1    2    [  A  S  P  N  ],   [  A  B  D  ]  =      1    2    [  A  B  C  D  ].For   △  A  M  R,  [  A  M  R  ]  =  [  A  B  C  D  ]  −  [  A  B  M  ]  −  [  C  R  M  ]  −  [  A  R  D  ]  =  [  A  B  C  D  ]  −      1    2    [  A  B  M  N  ]  −      1    2    [  C  R  P  M  ]  −      1    2    [  A  S  R  D  ]  =  [  A  B  C  D  ]  −            1      2        (  [  A  B  M  N  ]  +  [  C  R  P  M  ]  +  [  A  S  R  D  ]  )  =  [  A  B  C  D  ]  −            1      2        (  [  A  B  C  D  ]  +  [  A  S  P  N  ]  )  =      1    2    [  A  B  C  D  ]  −      1    2    [  A  S  P  N  ]  =  [  A  B  D  ]  −  [  A  S  N  ]

ABCD is a parallelogram, P is any point on AC. Through P, MN is drawn parallel to BA cutting BC in

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