mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 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.
Nice post! Just a minor nitpick, I found it weird that you say "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."
This is true but it's not needed (it's automatically true), you have in fact already proved that this is a parallelogram, by proving that two opposite sides have same length and are parallel (If you prove that the vectors EF and GH have the same coordinates, then EFHG is a parallelogram.)
Vivien
   ×1              Reply
mscroggs.co.uk is interesting as far as MATHEMATICS IS CONCERNED!
DEB JYOTI MITRA
                 Reply
 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 "o" then "d" then "d" in the box below (case sensitive):

Archive

Show me a random blog post
 2021 

May 2021

Close encounters of the second kind

Jan 2021

Christmas (2020) is over
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2021