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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.
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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]
∴ PHBE is a parallelogram
BE = PH ....(i)
⇒ AEPG is parallelogram
∴ AE = GP ....(ii)
But is the midpoint of AB
∴ AE = BE ...(iii)
from (i), (ii) and (iii)
⇒ GP = PH
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