Algebraic Proof: Diagonals of a Parallelogram Bisect Each Other

Your math teacher probably taught you proving theorems in Geometry using two-column proofs.  If your teacher has only taught you this type of proof, you may also use other proof techniques — I’m pretty sure he will give you full mark as long as your proof is correct.

When you go to college, most of the proofs that you will write is not in two columns but in paragraph form.  There are also times that you will use Algebra to prove Geometry problems or vice versa.  In this post, we do the former. We use Algebra to prove a theorem in Geometry.

The theorem below is about the diagonal of a parallelogram We will learn that the diagonals of a parallelogram bisect each other. The proof requires knowledge of the midpoint formula which we have already discussed earlier.

Theorem

The diagonals of a parallelogram bisect each other.

Proof

Construct a parallelogram OPQR where O is at the origin as shown below. If we let the coordinates of R be (a,0) and the coordinates of P be  (b,c), then the coordinates of Q are (a + b,c). Can you see why?

parallelogram-diagonal

To prove that \overline{PR} and \overline{OQ} bisect each other, we need to show that the diagonals have the same midpoint.

By the midpoint formula, the midpoint of \overline{PR} has coordinates (\frac{a+b}{2} , \frac{c}{2}).

Similarly, the midpoint of \overline{OQ} has coordinates (\frac{a+b}{2}, \frac{c}{2})

Since the midpoint of the two diagonals are equal, the theorem is proved.  \blacksquare.

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