Prove that a median divides a triangle into two triangles of equal area.

Let AD is a median of ∆ABC and D is the midpoint of BC. AD divides ∆ABC in two triangles: ∆ABD and ∆ADC.
To prove: ar(∆ABD) = ar(∆ADC)
Construction: Draw AL ⊥ BC.
Proof:
Since D is the midpoint of BC, we have:
BD = DC

Multiplying with $\frac{1}{2} A L$ on both sides, we get:

$\frac{1}{2} \times B D \times A L=\frac{1}{2} \times D C \times A L$

⇒ ar(∆ABD) = ar(∆ADC)