In a triangle ABC (see figure), E is the midpoint of median AD, show that

(i) ar ∆ABE = ar∆ACE

(ii) ar ∆ABE = ar(∆ABC)

i) Consider ΔABC

AD is the median which will divide area(ΔABC) in two equal parts

⇒ area(ΔABD) = area(ΔADC) …(i)

Consider ΔEBC

ED is the median which will divide area(ΔEBC) in two equal parts

⇒ area(ΔEBD) = area(ΔEDC) …(ii)

Subtract equation (ii) from (i) i.e perform equation (i) – equation (ii)

⇒ area(ΔABD) - area(ΔEBD) = area(ΔADC) - area(ΔEDC)

⇒ area(ΔABE) = area(ΔACE) …(iii)

ii) consider ΔABD

BE is the median which will divide area(ΔABD) in two equal parts

⇒ area(ΔEBD) = area(ΔABE) …(iv)

Using equation (iv), (iii) and (ii) we can say that

area(ΔABE) = area(ΔEBD) = area(ΔEDC) = area(ΔACE) …(v)

from figure

⇒ area(ΔABC) = area(ΔABE) + area(ΔEBD) + area(ΔEDC) + area(ΔACE)

using (v)

⇒ area(ΔABC) = area(ΔABE) + area(ΔABE) + area(ΔABE) + area(ΔABE)

⇒ area(ΔABC) = 4 × area(ΔABE)

⇒ area(ΔABE) = × area(ΔABC)