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.
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 "jump" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy τ+e-8 Approximation Day!

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

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

Archive

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