mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

A surprising fact about quadrilaterals

 2020-05-15 
This is a post I wrote for The Aperiodical's Big Lock-Down Math-Off. You can vote for (or against) me here until 9am on Sunday...
Recently, I came across a surprising fact: if you take any quadrilateral and join the midpoints of its sides, then you will form a parallelogram.
The blue quadrilaterals are all parallelograms.
The first thing I thought when I read this was: "oooh, that's neat." The second thing I thought was: "why?" It's not too difficult to show why this is true; you might like to pause here and try to work out why yourself before reading on...
To show why this is true, I started by letting \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) be the position vectors of the vertices of our quadrilateral. The position vectors of the midpoints of the edges are the averages of the position vectors of the two ends of the edge, as shown below.
The position vectors of the corners and the midpoints of the edges.
We want to show that the orange and blue vectors below are equal (as this is true of opposite sides of a parallelogram).
We can work these vectors out: the orange vector is$$\frac{\mathbf{d}+\mathbf{a}}2-\frac{\mathbf{a}+\mathbf{b}}2=\frac{\mathbf{d}-\mathbf{b}}2,$$ and the blue vector is$$\frac{\mathbf{c}+\mathbf{d}}2-\frac{\mathbf{b}+\mathbf{c}}2=\frac{\mathbf{d}-\mathbf{b}}2.$$
In the same way, we can show that the other two vectors that make up the inner quadrilateral are equal, and so the inner quadrilateral is a parallelogram.

Going backwards

Even though I now saw why the surprising fact was true, my wondering was not over. I started to think about going backwards.
It's easy to see that if the outer quadrilateral is a square, then the inner quadrilateral will also be a square.
If the outer quadrilateral is a square, then the inner quadrilateral is also a square.
It's less obvious if the reverse is true: if the inner quadrilateral is a square, must the outer quadrilateral also be a square? At first, I thought this felt likely to be true, but after a bit of playing around, I found that there are many non-square quadrilaterals whose inner quadrilaterals are squares. Here are a few:
A kite, a trapezium, a delta kite, an irregular quadrilateral and a cross-quadrilateral whose innner quadrilaterals are all a square.
There are in fact infinitely many quadrilaterals whose inner quadrilateral is a square. You can explore them in this Geogebra applet by dragging around the blue point:
As you drag the point around, you may notice that you can't get the outer quadrilateral to be a non-square rectangle (or even a non-square parallelogram). I'll leave you to figure out why not...

Similar posts

Mathsteroids
Interesting tautologies
Big Internet Math-Off stickers 2019
Runge's Phenomenon

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "m" then "e" then "d" then "i" then "a" then "n" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

Mar 2020

Log-scaled axes

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

trigonometry christmas frobel chebyshev radio 4 map projections inverse matrices chess data sound noughts and crosses preconditioning convergence numerical analysis countdown geogebra signorini conditions graph theory bempp weak imposition gaussian elimination determinants the aperiodical london underground coins manchester exponential growth matrix of cofactors talking maths in public captain scarlet big internet math-off speed cambridge error bars palindromes rhombicuboctahedron flexagons statistics logs pizza cutting games gerry anderson world cup people maths football stickers quadrilaterals sport wave scattering news ucl dragon curves manchester science festival martin gardner cross stitch final fantasy sorting game of life matrix of minors latex python propositional calculus hannah fry data visualisation bubble bobble phd misleading statistics matrix multiplication wool royal baby squares european cup harriss spiral estimation video games books accuracy go triangles a gamut of games javascript approximation hats inline code twitter dataset tmip platonic solids binary light php royal institution bodmas computational complexity london rugby fractals simultaneous equations puzzles pythagoras matrices curvature electromagnetic field arithmetic folding paper ternary tennis interpolation matt parker polynomials christmas card plastic ratio mathsjam menace graphs chalkdust magazine logic reuleaux polygons golden ratio reddit game show probability mathsteroids pac-man craft hexapawn dates finite element method folding tube maps raspberry pi golden spiral oeis realhats nine men's morris draughts geometry weather station mathslogicbot machine learning national lottery boundary element methods asteroids braiding advent calendar programming probability sobolev spaces

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020