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Constructions
Question
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In figure, E is any point on median AD of a ΔABC. Show that ar(ΔABE) = ar(ΔACE).

Answer

Given: E is any point on median AD of a ΔABC.
To Prove: ar(ΔABE) = ar(ΔACE).
Proof: In ΔABC,
∵    AD is a median. ∴ ar(ΔABD) = ar(ΔACD)    ...(1)
∵ A median of a triangle divides it into two triangles of equal areas
In ΔEBC,
∵    ED is a median.
∴ ar(ΔEBD) = ar(ΔECD)    ...(2)
∵ A median of a triangle divides it into two triangles of equal areas Subtracting (2) from (1), we get ar(ΔABD) – ar(ΔEBD)
= ar(ΔACD) – ar(ΔECD)
⇒ ar(ΔABE) = ar(ΔACE).

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