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Question

In the adjoining figure, D and E are points on the sides AB and AC of ABC such that ar(∆BCE) = ar(∆BCD). Show that DE || BC.

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Solution



∆BEC and ​∆BCD lies on the same base BC and have equal areas.
Darw EM as the altitude of ∆BEC and DL as the altitude of ∆BCD.
ar(∆BCE) = 12 ⨯ base ⨯ height = 12 ⨯ BC ⨯EM
ar(∆BCD) = 12 ⨯ base ⨯ height = 12 ⨯ BC ⨯DL
However, ar(∆BCE) = ar(∆BCD)
12 ⨯ BC ⨯EM = 12 ⨯ BC ⨯DL
⇒ ​EM = DL
i.e., the corresponding altitudes of both the triangles are equal.
∴ DE || BC

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