Euclidean Geometry for Maths Competitions
Geoff Smith
1/6/2015

In many cultures, the ancient Greek notion of organizing geometry into a deductive
system was taught using Euclid’s Elements, and the cultural consequences of this persist
to this day.

Euclid is not a model of perfection
Euclid organized a body of knowledge concerning plane geometry very well, and set up
an axiom system. He was sufficiently clear sighted to realise that he had no way of
deducing the ‘parallel postulate’ from the other axioms.
However, by the standards of modern mathematics, Euclid’s system looks very shaky.
We now use the language of sets and maps to express mathematics with exquisite
precision. Euclid had no such luxury, and is very vague about the meaning of “point”
and “line”.

Geometry is not uniquely suited to deductive reasoning
In the modern era, every branch of pure mathematics is a formal deductive system,and plane geometry has no special place, except for the historical accident that the firstattempts to use the axiomatic method were made in that context.

Where does one start?
Well, it is possible to develop Euclidean Geometry in a very formal way, starting withthe axioms. Some people advocate this as being a necessary part of education.
My personal opinion is that, for most people, this is not the sensible thing to do.
Now, this is only a personal opinion, and some people would disagree strongly, but I
will explain my attitude.
I think that the foundations of Euclidean Geometry, studied axiomatically, are rather
tricky, and the need to make sure that the “House of Cards” is being built correctly
actually distracts from the enjoyment of the subject.
I prefer that one regards it as intuitively clear what one means by “point”, “line”,“angle”, “length” and “area”. I think that it is also psychologically useful to accept
basic angle facts without question: vertically opposite angles are equal, corresponding
angles associated to a line transverse to a pair of parallel lines are equal. Simply accept
that SSS, SAS, ASA and AAS(corresponding) are legitimate justifications for triangles
to be congruent. Also accept that AAA and other related conditions are enough tojustify the similarity of triangles (for example, two triangles involve the same angle,
and the adjacent sides are in the same ratio).

These ideas could be analyzed carefully of course, and chased back to more primitivegeometric notions. It is important for some people to do this, because they cannot abide
informal foundations. However, from this position, I think that most people prefer tobuild palaces rather than dig to check that the foundations are solid. I am happy
for other people to check the foundations, and to enjoy myself playing with the fancy
architecture.

What should I learn?
Well, the best way to learn geometry is to do it. This means solving geometry problems.
However, there are a range of standard theorems which are appropriate to different levelsof mathematics competition. Of course you will need to know the basic “circle theorems”(angle in the alternate segment, angle subtended by an arc in a circle is half the angle
subtended at the centre, angle in the same segment etc) and the theorem of Pythagoras.
You should also understand the intersecting chords (and intersecting secants) theorem(also known as “power of a point”). On top of that, you should know at least a little
trigonometry, certainly the sine rule (a/ sin A = 2R) and the cosine rule.
You will need to learn results about isosceles triangles, equilateral triangles, parallelograms, trapezia (trapezoids), cyclic quadrilaterals.
Then there some more advanced theorems which are not necessary for very elementary competitions, but are essential for international competitions. These results
certainly include the theory of the Simson line and the Euler line, the theory of excircles
and Ptolemy’s theorem.
After that, it depends on your level of enthusiasm. You are likely to enjoy thefamous theorems of projective geometry: Pascal’s hexagram theorem, the theorem ofDesargues, the theorem of Brianchon. If your interest is strictly practical (I want anIMO medal) then you do not have to know the proofs of these fantastic results. You
can simply regard them as extra axioms of geometry and don’t tell people your guilty
secret that you have no idea why they are true.
You may want to supplement your Euclidean geometry techniques by becoming
confident in one or more of the various algebraic systems which can be used to resolve
geometric questions: trigonometry, complex numbers, vectors, areal co-ordinates (or thealmost equivalent trilinear co-ordinates). You might get interested in tiling methods, orproofs by the statics of mechanical systems, or proofs by origami. There are also many
formulas in the spirit of Heron’s formula which may be usefully learned.
You should definitely know Ceva’s theorem and the strange theorem of Menelaus.
This last result often enables us to show by calculation that three points are collinear.
Incidentally, I regard the use of Menelaus as involving an implicit reproach because itseems that when you find yourself using Menelaus, it means that you have missed abetter way to prove the collinearity. However, if the only way you can see to provea collinearity is to use Menelaus, then it is certainly the best method. It makes one
feel dirty of course, but it is much better than being stuck and unable to prove thecollinearity.
However, the algebra should be regarded as a bonus method. Most geometry questions found in maths competitions have (intended) solutions based on classic Euclideantechniques. Being confident at algebraic methods gives you more lines of attack if you
don’t see a Euclidean proof. However, simply learning the algebraic machines in orderto have an exhaustive knowledge of methods of attack is like purchasing lots of expensive
sports cars and keeping them in your garage. Most of the time, the fast solution is by
Euclidean methods, and algebra plays only a supplementary role, if any. The truth is,similar triangles and ingenuity are enough to solve most maths competition problems.

British Books
You should look for good material written in your language. By chance I happen tobe British. My maths enrichment organization is UKMT. It has a publishing arm
which currently produces three geometry books which are very relevant to competition
geometry. See http://www.ukmt.org.uk/publications
Bradley and Gardiner’s Plane Euclidean Geometry has seven chapters. I recommend
starting with Chapter 3. The first two chapters concern digging for the foundations. Of
course, that might appeal to you, but it is not necessary for competition mathematics.
The two books by Gerry Leversha are “Crossing the Bridge” (a not very scrutablereference to the Pons Asinorum), and the more sensibly titled “Geometry of the Triangle”.
If you come from a developed country, it is quite possible that you will have anational mathematics enrichment organization, and it may publish materials.

Local books and illegal books
If you come from a developing country, then the cost of books priced in hard currency
may be prohibitive, and it is less likely (but not impossible) that you will have a goodnational mathematics enrichment organization. Look to see if there are good materials
available produced in your own country. Also Kiran Kedlaya’s wonderful “GeometryUnbound” has been made freely available by the author (find the PDF using a searchengine).
Of course there are illegal copies of classic geometry books on the internet. If youcome from a developed country, I hope that you realise that it is illegal, immoral andcorrosive to steal intellectual property. I make no judgement on people from poor
countries who download such materials, for their situation is very different.

How should I study?
Solving geometry problems is easily the best way to become a good geometer. Simply
reading the theory will not do it. You have to engage directly with problems.
It is really, really important to draw good diagrams. This is because a clean, accurately drawn diagram is very likely to give you clues as to what is true. It will reveal
apparent collinearities of points, concurrencies of lines, and sometimes the fact thatfour points lie on a circle. Such help is invaluable. Time spent drawing a good diagram(using a sharp pencil, ruler and compasses) is not wasted. It is simply an investment
which will often quickly reward you with clues about how to solve the problem. It is avery good idea to practise drawing diagrams accurately (and eventually, quickly).