Constructing a Square

Date: 12/25/98 at 10:29:00
From: Larry Poleshuck
Subject: Construct a square

I found this problem in my high school geometry book 30 years ago and 
in all that time have not been able to solve it or prove it to be 
impossible.

"Given any four points, construct a square such that each side or 
extension passes through one point."

My teacher couldn't solve it and was unsuccessful in getting the answer 
from the author. I have long suspected this was a typo and that they 
really wanted a construction of a rectangle given any four points - a 
much easier task.

Can you help?

Date: 12/27/98 at 04:26:29
From: Doctor Floor
Subject: Re: Construct a square

Hello Larry,

Thanks for your question!

Let four points A, B, C and D be given. Then it _is_ possible to 
construct a square, such that each side or extension passes through 
one of the points:

 

Here's how:

Construct the midpoints E of AB and F of CD. These are the midpoints 
of the circles with diameters AB and CD. Draw these circles.

After this, draw the line through E perpendicular to AB. Let this line 
meet the circle in points I and J. Construct points L and K on the 
other circle in the same way.

Now take I or J on the first circle, and L or K on the other: for 
instance J and K. The choice must be made in such a way that the line 
JK intersects both circles in two points. So we find two more points 
(G and H) and these two points are the opposite vertices of the square 
we look for!

A word on why this construction works:

When we take any point X on the circumference of the AB-circle, we know 
that angles BXJ and JXA are half of angles BEJ and JEA: 45 degrees. We 
also know that angle BXA is 90 degrees. So XJ could very well serve as 
a diagonal of a square passing through A and B. In the same way, if we 
take any point Y on the circumference of the CD-circle, then YK can be 
a diagonal of a square with sides passing through C and D. By drawing 
JK and using the other two intersections G and H with the circles, we 
let the two diagonals 'XJ' (HJ) and 'YK' (GK) coincide. And then all 
fits into a square!

I hope this helped - if you have a question, write us back.

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/