Vector Proofs (Triangle and Parallelogram)

[kidluff]

Using Vectors, proof:

1)In triangle ABC, the midpoints of AB, BC, AC are DEF respectively. Prove a+b+c = d+e+f (these are position vectors).

2)Prove that if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

 

[soroban]

Hello, kidluff!

2) Prove that if the diagonals of a parallelogram are perpendicular,
then the parallelogram is a rhombus.

         A               B
          * - - - - - - *
         /             /
        /             /          Draw diagonals AC and BD.
       /             /
      * - - - - - - *
     D               C

We are told that AC is perp. to BD . . . that is: .AC · BD .= .0

We have: .AC .= .AB + BC
. . . . and: .BD .= .BC + CD .= .BC - AB

Then: .(AB + BC) · (BC - AB) .= .0

. . AB·BC - AB·AB + BC·BC - AB·BC .= .0

. . . . . BC·BC - AB·AB = 0

. . . . . . . . . . .AB·AB .= .BC·BC

. . . . . . . . . . . |AB|² .= .|BC|²

Therefore: . . . |AB| .= .|BC|

The parallelgram is a rhombus.