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Question
If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then Prove that `vec"AB" + vec"AD" + vec"CB" + vec"CD" = 4vec"EF"`
Solution
Given that ABCD is a quadrilateral.
E and F are the midpoints of AC and BD.
Let `vec"a", vec"b", vec"c"` and `vec"d"` be the position vectors of the vertices A, B, C and D respectively.
`vec"OA" = vec"a"`
`vec"OB" = vec"b"`
`vec"OC" = vec"c"`
and `vec"OD" = vec"d"`
E is the midpoint of AC.
∴ `vec"OE" = (vec"OA" + vec"OC")/2`
`vec"OE" = (vec"a" + vec"c")/2`
F is the midpoint of BD.
∴ `vec"OF" = (vec"OB" + vec"OD")/2`
`vec"OF" = (vec"b" + vec"d")/2`
`vec"AB" + vec"AD" + vec"CB" + vec"CD" = vec"OB" - vec"OA" + vec"OD" - vec"OA" + vec"OB" - vec"OC" + vec"OD" - vec"OC"`
= `2vec"OB" - 2vec"OA" + 2vec"OD" - 2vec"OC"`
= `2vec"b" - 2vec"a" + 2vec"d" - 2vec"c"`
`vec"AB" + vec"AD" + vec"CB" + vec"CD" = 2(vec"b" - 2vec"a" + vec"d" - vec"c")` ......(1)
`vec"EF" = vec"OF" - vec"OE"`
= `(vec"b" + vec"d")/2 - (vec"a" + vec"c")/2`
= `(vec"b" + vec"d"- vec"a" + vec"c")/2`
`vec"EF" = (vec"b" - vec"a" + vec"d" - vec"c")/2`
`2vec"EF" = vec"b" - vec"a" + vec"d" - vec"c"` .......(2)
Fro equation (1) and (2) we have
`vec"AB" + vec"AD" + vec"CB" + vec"CD" = 2(2vec"EF")`
`vec"AB" + vec"AD" + vec"CB" + vec"CD" = 4vec"EF"`
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