Given : ΔABC, P is the mid point of AB and PQ BC Prove that : Q is the mid point of AC. 


1) P is the mid point  1) Given (ΔABC) 
2) AP = PB  2) Definition of mid point 
3) AP / PB = 1  3) Ratio of two equal side is 1. 
4) PQ  BC  4) Given (ΔABC) 
5) ( AP / PB ) = ( AQ / QC )  5) Basic Proportionality Theorem 
6) AQ / QC = 1  6) From (3) and (5) (Transitivity) 
7) AQ = QC  7) Cross multiply 
8) Q is the mid point of AC  8) Definition of mid point 
Given : 1) ABCD is a trapezoid. 2) AB  DC Prove that : OE  DC. Construction : Draw OE such that OE DC 


1) OE  DC  1) Construction (ΔABC) 
2) (BO / OD) = ( BE / EC)  2) Basic Proportionality Theorem 
3) AB  DC  3) Given 
4) OE  AB  4) From (1) and (3) (Transitivity) 
5) (AO / CO) = (BE / EC)  5) Basic Proportionality Theorem 
6) (AO / CO) = (BO / DO)  6) From (2) and (5) ( Transitivity) 
7) (AO / BO) = (CO / DO)  7) From above ( Alternendo) 