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.
 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 "u" then "n" then "c" then "o" then "u" then "n" then "t" then "a" then "b" then "l" then "e" 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

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

Archive

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