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Question
ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH
Solution
Since E and F are midpoints of AB and CD respectively
∴ AE = BE =`1/2` AB
And CF = DF =`1/2` CD
But, AB = CD
∴ `1/2` AB = `1/2` CD
⇒ BE = CF
Also, BE || CF [∵AB || CD]
∴ BEFC is a parallelogram
⇒ BC || EF and BF = PH ....(i )
Now, BC || EF
⇒ AD || EF [ ∵ BC || AD as ABCD is a parallel]
⇒ AEFD is parallelogram
⇒ AE = GP
But is the midpoint of AB
∴ AE = BE
⇒ GP = PH
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