Triangles on the same base and between same parallels are equal in area.
: Two triangles ABC and DBC are on the same base BC and between same parallels EF and BC.
To Prove : ar(ΔABC) = ar(ΔDBC)
Construction : Through B, draw BE || AC intersecting line AD in E and through C draw CF || BD intersecting the line DA in F.
Proof : EACB and DFCB are parallelograms (Since two pairs of opposite sides are parallel)
Also ||gm EACB and ||gm DFCB are on the same base BC and between same parallels EF and BC.
∴ ar(||gm EACB)= ar(||gm DFCB)...(i)
Now AB is the diagonal of ||gm EACB
∴ ar(ΔEAB) = ar(ΔABC)
∴ ar(ΔABC) = (1/2)ar(||gm EACB) .....(ii)
ar(ΔDBC) = (1/2) ar(||gm DFCB)...(iii)
From equations (i), (ii) and (iii), we get
ar(Δ ABC) = ar(Δ DBC)
See the figure