Squares on a Quadrilateral

Tuesday, September 1st, 2020

A recent Quora question asks the following:

Squares are constructed externally on the sides of an arbitrary quadrilateral. How do you show that the line segments joining the centers of opposite squares lie on perpendicular lines and are of equal length?

I chose to answer it because it provides a nice opportunity both to use what I like to call “continuous induction” and to illustrate the proof with an interactive diagram.

The result is clearly true for a square. So it will suffice to show that if we have any quadrilateral for which it is true and move just one corner then it will be true for the new quadrilateral. A picture would help, but I think you will enjoy (and learn more from) the effort of making your own rather than just scrolling down to look at mine.

I found it helpful to use vector language for describing the points involved, but there is also a nice geometric version of the argument that you might enjoy looking for before scrolling down.

Let the quadrilateral be ABCD and consider the effect of moving B by some displacement vector v. Let’s use s for the side AB and t for the side BC. Then the centre of the square on AB is displaced from A by the vector (s+Rs)/2 where R is a rotation by Pi/2 (counterclockwise if ABCD represents a clockwise orientation of the corners of the quadrilateral), and the centre of the square on BC is displaced from A by s+(t+Rt)/2.

After moving B by the vector v, the vector s is replaced by s+v and t by t-v.

So the centre of the AB square is moved by ((s+v)+R(s+v))/2 -(s+Rs)/2=(v+Rv)/2

and for the BC square it’s moved by [s+v+{(t-v)+R(t-v)}/2]-[s+(t+Rt)/2]=(v-Rv)/2

But since R^2=-1 (two right angle rotations give a reversal), R(v-Rv)/2=(v+Rv)/2

So the motions of the centres are perpendicular and of equal length (and the direction of the rotation from one displacement to the other is the same as that from one midpoint line to the other)

But if the lines joining the opposite pairs of centres are perpendicular and of equal length, and the displacements of the new centres are also perpendicular and of equal length, then (with the directions of rotation matching) the triangles formed by old and new centre lines and their endpoint displacements are congruent and so the lengths of the new centre lines are also of equal length and the angle between them is the same as between the old ones.

If you drew a picture to follow along with the above it probably looked something like this:

But I’ve now made a better version:

which also has the advantage of showing a more intuitive and geometrical version of the argument. So the promised shorter proof goes as follows:

When a corner B is moved to a new position B1, the lines joining its counterclockwise neighbour A to the old and new centres S and S1 are rotated counterclockwise by 45 degrees from the corresponding old and new edges, and scaled by a factor of 1/root(2). So the displacement SS1 is similarly related to BB1. And by the same argument the displacement TT1 of the centre of the square on the neighbouring side in the other direction is also scaled by a factor of 1/root(2) and rotated by 45 degrees in the clockwise direction. So SS1 and TT1 are perpendicular and of equal length. So if we start with a case where the midpoint-joining lines US and VT are perpendicular and of equal length, then by sAs congruence we see that US1 and VT1 are also equal and perpendicular. But, as noted at the outset, the result is obvious for a square and any other quadrilateral can be reached from a square by just moving vertices one-by-one.

Diigo Alternative

Monday, May 25th, 2020


What is a Fraction?

Wednesday, September 26th, 2012

A couple of Calgary Math Ed students have announced on LinkedIn that they are starting a Concept Study of Fractions.

This is a good idea as the topic is often challenging for students and I suspect that one reason lies in its language.
“A fraction is part of a whole” is consistent with the use in chemistry and with the concept of “proper fraction” sometimes introduced in math classes. Some would interpret “fraction” as equivalent to “ratio” (including ratios of irrationals) but others restrict to ratios of integers and some of these would include 3/2 as a fraction but not 6/3 (with the idea that a fraction is a rational number that is not an integer). And yet others would deny that a fraction is a number at all – with the word “fraction” referring to a way of naming a number rather than the number itself.

The difficulty of a not insignificant jump in the understanding of quantity is compounded by seeing all these various usages at the same time.

(and not even just from different sources – many teachers are themselves confused enough to keep switching from one to another in the course of essentially the same discussion)

On the other hand the desire for precision can lead to an apparent arbitrariness which is also off-putting to students. So I suspect that another major problem is the fact that some teachers, in their justified zeal to emphasize the need for precise definitions, often fail to emphasize the equally important issue of *scope* of a definition.

Are You Ready for Calculus?

Tuesday, September 4th, 2012

If you are not sure whether you need a precalculus course before going on to calculus, you may want to look at some of the links on our precalculus index page.

And if you are not sure which of the many topics are most important, read on.

In 1999, the Centre for Curriculum Transfer and Technology in BC completed a project to review the general and specific mathematics proficiencies recommended by BC’s post-secondary educational institutions for entry to selected mathematics/statistics courses. The courses selected for this study were

  • Calculus,
  • Introductory Statistics, and
  • Mathematics for Elementary Education.

The Report details some very interesting findings and details the recommendations of the Project’s Steering Committee.

Perhaps the most significant findings were these:
! Students are expected to have strong reading and writing skills in order to succeed in mathematics.
! Working with lines on a Cartesian graph is a most important skill.
! The top Proficiency Category is M (The Straight Line and Linear Functions).
! Understanding and using the slope of a line was ranked the overall most important mathematical proficiency for both Calculus AND Statistics.

You can download the Report from this link (you’ll need a pdf reader).

More Calculators

Tuesday, September 4th, 2012

In addition to general purpose calculators that are built in to any decent smartphone, and web-based tools like Wolfram Alpha and various other on-line graphing utilities, there are also sites devoted to specific topics which have built-in formulas for calculating anything imaginable – from mortgage payments to gas mileage to calorie counting.

One massive collection is Martindale’s.

Another, which was recommended recently by students of Jessica Lee at coloradotutors.org ,is this collection which has some which may be fun for students and which also includes a bit of advice on using such things.

Desmos Graphing Calculator

Wednesday, August 1st, 2012

This seems to implement almost all of the features of my old Graph Explorer with the addition of a gui math editor and an overall more attractive look that I won’t be able to match.

I may start using it myself for graphing (at least when I don’t need to control parameters to arbitrary precision), but will keep using the GeX library for development of mathlets on specific topics.

Thanks to Colleen Young for the link

Which is easier to teach and understand – fractions or negative numbers?

Wednesday, July 25th, 2012

Erlina Ronda is a Math Ed specialist at the University of the Philippines who seems seriously interested in teaching for understanding and her latest blog post asks Which is easier to teach and understand – fractions or negative numbers?

Interesting question!  But I suspect that it doesn’t have an answer as some people may be better conditioned (either by make-up or experience) to handle one, and others may find it easier to deal with the other.


With regard to fractions I think one of the main barriers, in addition to the fact that the notation is more complex (each fraction being made up out of two other numbers), is the non-uniqueness of representation – by which I mean the fact that two different fractions (eg 4/6 and 2/3 ) can represent the same value.

And with regard to signed numbers I think it is partly (as Erlina noted) breaking away from the concept of numbers as representing just some kind of quantity or size, and letting them also have a direction.  In fact the signed numbers (despite having the “advantage” of a more unique symbol) are better thought of as relationships (translations) than values (positions on the line).  So +5 is an increase of 5 units   and -3 is a decrease of 3 units (each of which corresponds to a whole family of differences just as each rational number corresponds to a whole family of divisions)

This approach makes it easier to understand the commerce-based rules for signed multiplication that were laid out by Brahmagupta in 7th century India. It may not be so obvious what was meant by the “product” in what is often translated as  “The product of two debts is a fortune” , but read as “the loss of a loss is a gain” it becomes much more obvious.


I guess I could say that signed numbers and fractions are both about relationships between pairs of magnitudes  – or more properly families of pairs that are all related in the same way (and when people come back to elementary arithmetic in advanced math classes that is really how they do it).

The problem with fractions then is mainly that this is more explicit in the notation (so the complication is visible) and the problem with negatives being that it is not (so something important is hidden)!




Quadratic Functions

Wednesday, May 18th, 2011

Murray Bourne of squareCircleZ has posted on ‘How to find the equation of a quadratic function from its graph‘. This is indeed the type of discussion and exercise that we need to see more of.  Not only does it promote a deeper understanding of the mathematics than the reverse but it is also relevant to more practical applications. Occasionally we do come up with a formula and want to see what it looks like but, especially when it comes to specific examples as opposed to general patterns, it is more often that we have data and want to find or verify a formula.  One of the activities in my own “Blue Meanies” game (at http://qpr.ca/math/applets/meanies/ )asks students to “guess” the equation of a parabola through three points by imagining the curve and using its geometry (in various ways) to determine the equation. Of course in such “modelling” problems, with limited data there will be many possible model types that can be used, and there is an interesting interplay between fitting with a particular class of functions (eg polynomial or exponential) and giving reasons why one or other such class might be more appropriate in a given situation.


SquareCircleZ and IntMath

Monday, February 28th, 2011

Murray Bourne in Singapore provides both an interesting and enjoyable blog called squareCircleZ and a collection of  lessons on various topics called Interactive Mathematics “where you learn math by playing with it!”.

He frequently links to good examples of  free math resources, and also provides up-to-date evaluation of math rendering systems for the web.

Why a CMR Blog?

Thursday, July 29th, 2010

The goal here is to provide a vehicle for publicizing new resources and other changes at the CMR website.