=0 The figure below represents the triangle ABC and midpoints.
The area of the triangle ODE, AODE‌‌=‌
1
2
|OD×OE| ‌‌=‌
1
2
|(‌
a+c
2
)×(‌
b+c
2
)| ‌‌=‌
1
8
(a×b+a×c+c×b)‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(I) The given equation can be written as, a+2b+3c=0‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(II) Multiply b in equation (II) a×b+2(b×b)+3(c×b)=0 a×b+3(c×b)=0‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(III) Multiply c in equation (II). a×c+2(b×c)+3(c×c)=0 a×c+2(b×c)=0‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(IV) Add equation (III) and (IV) a×b+3(c×b)+a×c+2(b×c)=0 a×b+a×c+c×b=0 Substitute the above value in equation (I). AODE=0