Prove that the area of a triangle is half the product of any its sides and the corresponding altitudes .

Construction: Trough C and A draw CP parallel to BA and AP parallel to BC, intersecting each other at P.

Proof:

CP parallel to BA (By Construction)

AP parallel to BC (By Construction)

Therefore BCPA is a parallelogram.

Hence AC is a diagonal of parallelogram BCPA.

Therefore, area of triangle = ½ ar(II^{gm} BCPA) = ½ × BC × AD (As BC is the base and AD is the altitude of II^{gm} BCPA).

**
**