How to Draw a Parallelogram with One Ruler

Posted: March 8, 2018 in Mathematics
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Usually in school, you learn how to construct a parallelogram with compass and straightedge. But today, I am gonna show you how to construct a parallelogram with only one ruler. Of course, you need a piece of paper, a pencil, a hand, and your brain as well, etc …

Actually the method is very simple, even an eight year old can do it I promise. So here it is:

  1. Draw a quadrilateral (4-sided shape), any quadrilateral will do even the non-convex one.
  2. Find the midpoint of all four sides with your ruler.
  3. Connect those four midpoints together and you are done.

Here are some examples.

DrawingParallelogram

The proof is very simple.

Let the four points A(a_1, a_2), B(b_1, b_2), C(c_1, c_2), D(d_1, d_2) be the vertices of a quadrilateral.

DrawingParallelogramProof

Then the four midpoints are

\displaystyle P\left(\frac{a_1+b_1}{2}, \frac{a_2+b_2}{2}\right), Q\left(\frac{b_1+c_1}{2}, \frac{b_2+c_2}{2}\right), R\left(\frac{c_1+d_1}{2}, \frac{c_2+d_2}{2}\right), S\left(\frac{d_1+a_1}{2}, \frac{d_2+a_2}{2}\right)

Find the slopes of the sides and show that the opposite sides are parallel:

m_{PQ} = \dfrac{c_2-a_2}{c_1-a_1} = m_{RS} \Rightarrow PQ \;//\; RS

m_{QR} = \dfrac{d_2-b_2}{d_1-b_1} = m_{SP} \Rightarrow QR \;//\; SP

Therefore PQRS is a parallelogram.

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