Question

If E is any point on median AD of $△$ABC prove that $A\left(△ABE\right)=A\left(ACE\right)$.

Answer 1

$ABC$ is a triangle with $AD$ as median

i.e, $BD=CD$

$E$ is any point on $AD$

Join $EB$ and $EC$

In $\Delta ABC$

$D$ is the midpoint of $BC$

therefore $AD$ is median

since median divides a triangle a triangle into two triangle of equal area

$\therefore ar\left(\Delta ABD\right)=ar\left(\Delta ACD\right)⟶\left(1\right)$

similarly, In $\Delta EBC$

$D$ is the midpoint of $BC$

therefore $ED$ is the median

hence, $ar\left(\Delta EBD\right)=ar\left(\Delta ECD\right)⟶\left(2\right)$

subtracting (1) and (2)

$ar\left(\Delta ABD\right)-ar\left(\Delta EBD\right)=ar\left(\Delta ACD\right)-ar\left(\Delta ECD\right)\Rightarrow ar\left(\Delta ABE\right)=ar\left(\Delta ACE\right)$