Question

Any point D is taken on the side BC of a triangle ABC and AD is produced to E such that AD is equal to DE, prove that area of triangle BCE is equal to the area of triangle ABC.

Answer 1

AnswEr:

In ∆ ABE D is the mid-point of AE. Therefore, BD is the median. Since median divides a triangle in two triangles of equal area.

$\qquad\tt{\therefore\:ar(\triangle\:ABD)=ar(\triangle\:EBD)-\:-\:-\:(1)}$

• In ∆ ACE, D is the mid-point of AE

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$\qquad\tt\green{CD\:is\:median\:of\:AE}$

$\qquad\tt{\Rightarrow\:ar(\triangle\:ACD)=ar(\triangle\:ECD)-\:-\:-\:(2)}$

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Adding (1) and (2), we get

ar (∆ABD) + ar (∆ACD) = ar (∆EBD) + ar (∆ECD)

$\longrightarrow$ $\qquad\sf\blue{ar\:(\triangle\:ABC)=ar\:(\triangle\:BCE)}$