Question

in a triangle ABE,e is the midpoint of median ad prove that ar(BED=ar(aec).

Answer 1

since AD is the median of the triangle ABC , ar(ABD)=ar(AED)
since E is the midpoint of AD , in triangle ABD , BE is the median , ar(ABE)= ar(DBE)
since CE is the median  of triangle ACD , ar(ACE)=ar(DCE)
since area of the 2 triangles are equal , 
ar(BED)=ar(AEC)

Answer 2

in triangle (ABC),ad is a median

we know that a median divides a triangle into two traiangle of equal areas

hence (ABD) = (ACD)----------1

in triangle (BEC),ed is a median

we know that a median divides a triangle into two traiangle of equal areas

hence (BED) = (CED)----------2

eqn¹-eqb²

ar(ABD) - ar(BED) = ar(ACD) - ar(CED)

ar(EBD) = ar(ECD)

h.p