D is the midpoint of side BC of ΔABC and E is the midpoint of BD.
Question:

D is the midpoint of side BC of ΔABC and E is the midpoint of BD. If O is the midpoint of AE, Prove that ar(ΔBOE) = (1/8) ar(ΔABC).

Solution:

Given that

D is the midpoint of sides BC of triangle ABC

E is the midpoint of BD and O is the midpoint of AE

Since AD and AE are the medians of triangles, ABC and ABD respectively

∴ ar(ΔABD) = (1/2) ar(ΔABC)   ⋅⋅⋅⋅ (1)

∴ ar(ΔABE) = (1/2) ar(ΔABD)   ⋅⋅⋅ (2)

OB is the median of triangle ABE

Therefore, 

 ∴ ar(ΔBOE) = (1/2) ar(ΔABE)

From 1, 2 and 3, we have

∴ ar(ΔBOE) = (1/8) ar(ΔABC)

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