Length of Median of Triangle and Parallelogram Theorem

 

                     The following theorem is known as “Theorem of Parallelogram”.  After the proof of the theorem, we will introduce why this theorem got this name. 

 

 

Theorem (  Parallelogram )

                               =  .  Then

                                    ² +  ² = 4  ² +  ²

                  

                             

 

   Proof:

 

1.      From point A, construct a perpendicular line to .

Label the intersection point as P as shown in the diagram.

Then

           = ½ a + t

           = ½ a – t

 

2.      From Pythagorean theorem,

 

            r²= c² – ( ½ a + t )² = b² – (  ½ a –t )²

 

          c² – ( ¼ a² + at + t² ) = b² – ( ¼ a² – at + t² )

         t =

           Thus,

                       r² = b² –  ¼ a² + ½ ( c² – b² ) – t²

                     r² + t² = ½ b² + ½ c² – ¼ a²

 

3.      ² = r² + t²

                  = ½ b² + ½ c² – ¼ a²

 

        So,

                  ² +  ² = 4  ² +  ²

 

 

 

 

                   On , find a point D such that   =  as shown below:

                                 

 

Then   quadrilateral ABCD is a parallelogram.

From the theorem above,  we have

                      ² +  ² =  ² +  ²

 

And the result  is known as “Parallelogram Theorem”.