ABCD is a parallelogram .E is …

Given:
(1) ABCD is a parallelogram. 
(2) BE = 2EA
(3) DF = 2FC

To prove: AECF is a parallelogram. whose area is one third the area of parallelogram ABCD. 

Construction: Draw ALDC                  

Proof: ABCD is a parallelogram.
Therefore, ABDC
Since E and F are points on AB and DC respectively, 
AEFC 
AB = EA + BE
  = EA + 2EA  [BE = 2EA] 
  = 3EA 
 AE = AB  (i)                    
DC = DF + FC
 = 2FC + FC  [DF = 2 FC] 
 = 3FC 
 FC = DC  (ii)                      
Since AB = DC (parallel sides of the parallelogram), from (i) and (ii), we have, 
AE = FC 
Thus, AE = FC and AEFC
 AFEC
Therefore, AEFC is a parallelogram.                        
Now, since AECF and ABCD lie between the same parallel lines, their heights are also equal. 
Therefore, area (parallelogram AECF) = FC  AL 
 =  DC  AL
 = [area (parallelogram ABCD)]