The Butterfly Theorem

March 1, 2015

Let M be the midpoint of the chord PQ of a circle. Let AB and CD be two other chords of the circle passing through M and let X and Y be the intersection points of PQ with AD and BC respectively. Then M is the midpoint of the segment XY. Butterfly Theorem diagram (from wikipedia) Proof: The space of all degree two polynomials vanishing at the four points A, B, C and D is of dimension 2*. Thus it is spanned by the equation of the circle and the equation of the union of the lines AB and CD. Choose coordinates such that the line PQ is the x-axis and the point M is the origin. Then the coefficient of x in both our spanning polynomials is equal to zero, hence this coefficient is also zero in the equation of the union of AD and BC. Therefore M is the midpoint of the segment XY, as required. * To see this there is a six dimensional space of degree two polynomials and we are imposing one linear condition each time we require the polynomial to vanish at a point. These four conditions are linearly independent since it is easy to find degree two curves passing through three of these points and not the fourth (e.g. a union of two lines).

Links

February 11, 2015

Daniel discusses why what you see on your TV screen may be misleading. I wonder what the BCCI think of this.

An Oldie but a Goddie: Why South Africa has never won the f**ng World Cup.

On the Time Spent Preparing Grant Proposals. (hat-tip to SY).

The Dot and the Line (A Romance in Lower Mathematics) (youtube). This is actually a book, which I learnt on Saturday when I found the book in the Boundary Street Markets.

And last but not least, Australia will be competing in Eurovision 2015. And there is a petition calling on TISM to be our entry.

The Quest for the Perfect Pad See Eiw

February 4, 2015

That is, the quest for the perfect Pad See Eiw that is not located in Sydney. This quest has now entered its tenth year and we now discuss last night’s entry, from Thai Nakonlanna in St Lucia, Qld.

Unfortunately there will be no pictures, even though I had my camera with me. Sorry if you’re the type of person that actually thinks a picture is worth a thousand words.

Before we get to the food, I want to mention the cost. The Pad See Eiw with chicken here was 12AUD. As a point of comparison, Thai La-Ong charges 10.5AUD (dinnertime price). There’s this myth floating around that Sydney is the most expensive city in Australia, but after seeing prices here in Brisbane I’m calling Bravo Sierra on that one.

With much anticipation the first mouthful left the plate, was delicately transported upwards on a fork and entered the mouth. In an instant, I knew that this was not the one. It’s hard for me to describe exactly what was wrong in words, though I feel it was not sweet enough.

I don’t know why I bother continuing on this futile quest. I really should give up.

Ender’s Game

January 9, 2015

It was yonks and yonks ago that I read Orson Scott Card’s Ender’s Game and enjoyed the book so much that I happily continued with the sequels. It was with some trepidation however with which I approached the movie, as I had heard some indifferent reviews but also because of the following result:

Theorem: The book is always better than the movie.

(Proof: exercise left to the reader)

The movie follows exclusively the major plotline of the story. Thus it is essentially about Ender and Ender alone. It does at times skip through this storyline at a fast pace, sacrificing character development to do so which makes some of the scenes appear less meaningful or understandable when compared to the book version.

Disappointingly, the entire plot line involving Locke and Demosthenes is missing from the film. Thus if you’ve only seen the film, then you won’t get . The joke in is still understandable.

After watching the film, I felt I had to reread the book again to make a comparison between the two – as well as to test how accurate my memory of the book acually was. It felt weird reading dialogue which was inserted directly from the book into the film and thus I had just heard.

My recommendation is that the book is a must read. The movie is optional. Based on m y personal experiences with Jurassic Park and The Lost World (both of which I saw before I read), it is perhaps better to watch the movie before reading the book if you want to do both and for some reason haven’t yet read the book.

On the Translation of Poetry

December 20, 2014

OLYMPUS DIGITAL CAMERA

Taken at the Seoul Museum of Art.

Flamingos!

October 5, 2014

Taken in Colombia.

Sydney v Richmond

August 27, 2014

You are the captain of Sydney. The phone rings and you pick it up. Trent Cotchin is on the line.

“G’day Trent, what’s up?”

“I’m calling to discuss our game on this weekend. It seems that there’s an opportunity for us both to arrange things to benefit both of us.”

“Um, how?”

“Well what do you guys want to get out of this weekend’s match? The minor premiership, for sure, but even more importantly you want none of your players to get injured or suspended. Agree?”

“Yes”

“Well listen, I have a way to guarantee that you get all of the above. Interested?”

“Um, I’m not sure how you can guarantee that we don’t have any injuries.”

“All I ask for in return is that you let us win. Only by a small enough margin that you still keep top spot from the Hawks on percentage. But you see, we do really need the win more than you guys this time so we can make the eight.”

Trent pauses to take a breath, and continues

“Here’s how I propose we work things out. Come Saturday’s game, when the ball is bounced you allow us with no contest to score a couple of goals. Then both teams agree to sit down for the rest of the game and do nothing. We both get what we want out of the game without having to take any risks. Do you agree to this proposal?

What is your response?

(Also, if you accept, what will be the response of the rest of the world to such a highly contrived outcome?)

Concyclicity in the Euclidean Plane

August 21, 2014

I want to address the question of how to determine whether or not four points are concyclic. In a previous post, this question reared its head and was solved by an introduction of the circumcentre of three of the points. In this post, I present a direct approach.

We prove the following:

Let (x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4) be four points in \mathbb{R}^2. Then these four points are concyclic if and only if

\displaystyle \left| \begin{array}{cccc} x_1^2+y_1^2 & x_1 & y_1 & 1 \\ x_2^2+y_2^2 & x_2 & y_2 & 1 \\ x_3^2+y_3^2 & x_3 & y_3 & 1 \\ x_4^2+y_4^2 & x_4 & y_4 & 1 \end{array} \right|=0.

For the proof, first suppose that the points (x_i,y_i) all lie on the curve

\displaystyle (x-x_0)^2+(y-y_0)^2=r^2.

The four equations we obtain give a linear dependence of the columns in the 4\times 4 matrix under consideration, hence its determinant is zero.

Conversely, suppose that the determinant in question is zero. Then there is a nonzero vector in the nullspace of our 4\times 4 matrix. If its first entry is nonzero, we can normalise it to be of the form (1,-2x_0,-2y_0,x_0^2+y_0^2-r^2)^T and we obtain the circle that the four points lie on. If the first entry of this vector is equal to zero, then the four points must lie on a line, which after is all is just a degenerate circle, QED.

bartogian does the IMO (2014 Q4)

July 16, 2014

The fourth question on this year’s IMO is a Euclidean geometry question. And so I thought to myself, I wonder if I could get a computer algebra package (with Grobner bases, etc) to do the relevant algebraic manipulation to solve the problem for me, once I had converted the problem into coordinate geometry.

This is a desription of that journey. But first the problem:

Points P and Q lie on side BC of acute-angled triangle ABC such that \angle PAB=\angle BCA and \angle CAQ = \angle ABC. Points M and N lie on lines AP and AQ, respectively, such that P is the midpoint of AM and Q is the midpoint of AN. Prove that the lines BM and CN intersect on the circumcircle of triangle ABC.

As every good student knows, it is a good idea to do a little bit of Euclidean geometry before starting on the trigonometry or coordinate geometry. So we will begin by noticing that APQ is isosceles.

We will set up our coordinates so that the line BC is the horizontal axis, with the origin at the midpoint of PQ. Then we introduce variables h,x,y,z by A=(0,h), P=(x,0), Q=(-x,0), B=(y,0) and C=(z,0). Hence M=(2x,-h) and N=(-2x,-h).

Let X=(p,q) be the intersection of BM and CN and let O=((y+z)/2,w) be the circumcentre of ABC.

So now for the equations that these seven variables satisfy.

Since X lies on BM, there is

\displaystyle \frac{q}{p-y} = \frac{-h}{2x-y}

and since X also lies on CN, there is

\displaystyle \frac{q}{p-z} = \frac{-h}{-2x-z}

From |AO|=|BO| there is

\displaystyle \left(\frac{y+z}{2}\right)^2 + (w-h)^2=\left(\frac{y-z}{2}\right)^2+w^2

and from the similarity of triangles ABC and QAC, in particular AC^2=CQ\cdot BC, we get

\displaystyle h^2+z^2 = (z+x)(z-y).

And what do we have to prove? Well nothing other than |AO|=|XO|, so it suffices to prove

\displaystyle \left(p-\frac{y+z}{2} \right)^2+(q-w)^2 = \left(\frac{y+z}{2}\right)^2+(w-h)^2

So, we go to sage (I skip the routine and boring simplification) and execute

sage: x,h,y,z,w,p,q=QQ['x,h,y,z,w,p,q'].gens()
sage: I=ideal(q*(2*x-y)+h*(p-y),q*(2*x+z)-h*(p-z),y*z-2*w*h+h^2,h^2+z^2-(z+x)*(z-y))
sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in I
False

Yikes!

Well how about:

sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in I.radical()
False

Yikes again!

While it is tempting to expect a typographical error, that is not in fact the case. The following computation reveals what is really going on.

sage: I.associated_primes()
[Ideal (q, z, y, h) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (q, z, h, x) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (q, y, h, x) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (z, y, h, x) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field, Ideal (y*p + z*p - 2*p^2 - 2*h*q + 4*w*q - 2*q^2, y*z - p^2 - 2*h*q + 2*w*q - q^2, h*z - h*p + 2*x*q + z*q, 2*x*z - 2*h*w - 2*x*p - z*p + p^2 + h*q - 2*w*q + q^2, h*y - h*p - 2*x*q + y*q, 2*x*y + 2*h*w - 2*x*p - z*p + p^2 + h*q - 2*w*q + q^2, h^2 - 2*h*w + p^2 + 2*h*q - 2*w*q + q^2, z*p^2 - p^3 - 2*x*h*q + 4*x*w*q - h*p*q + 2*w*p*q - 2*x*q^2 - p*q^2, z^2*p - p^3 - 4*x*h*q + 8*x*w*q + 4*z*w*q - 2*h*p*q + 2*w*p*q - p*q^2, 2*h*w^2 - w*p^2 + 2*x^2*q - h*w*q + 2*w^2*q - w*q^2, h*w*p^2 - 2*x^2*h*q + 4*x^2*w*q - 2*x^2*q^2, w*p^4 - 4*x^2*h*w*q + 8*x^2*w^2*q - 2*x^2*p^2*q - 2*w^2*p^2*q + 2*x^2*h*q^2 - 8*x^2*w*q^2 + w*p^2*q^2 + 2*x^2*q^3) of Multivariate Polynomial Ring in x, h, y, z, w, p, q over Rational Field]

And we see the issue. The ideal I has five associated primes, but only one of these primes corresponds to the geometric problem at hand, the other four correspond to highly degenerate triangles.

So we try

sage: J=I.associated_primes()[4]
sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in J
True

and the proof is complete.

An alternative approach would be to artificially invert h by creating a new variable t satisfying the relation ht=1:

sage: x,h,y,z,w,p,q,t=QQ['x,h,y,z,w,p,q,t'].gens()
sage: I=ideal(q*(2*x-y)+h*(p-y),q*(2*x+z)-h*(p-z),y*z-2*w*h+h^2,h^2+z^2-(z+x)*(z-y),h*t-1)
sage: (p^2-p*y-p*z+(q-w)^2-(w-h)^2) in I
True

and QED for the second time.

Bartogian’s guide to Seattle

May 3, 2014

Do: Underground Tour. I’ve only done the day tour and think night tour is essentially the same but with more emphasis on the history deemed inappropriate by the morality police (eg, prostitution). The ticket can give you some discounts on other touristy things.

Eat: Mod Pizza. Good pizza at a good price. Right across the road from WAC.

Sing: Hula Hula. Karaoke every evening. Was not crowded on a Thursday evening, can’t speak for other days of the week.


Follow

Get every new post delivered to your Inbox.