Toronto Math Forum
MAT4752017S => Topics => Topic started by: Michael Chow on January 18, 2017, 07:07:46 PM

Hi everyone,
I will be giving a presentation on a topic in Euclidean geometry  Collinearity and Concurrency.
In my experience, students in Canada get $\varepsilon$ exposure to Euclidean geometry in both high school and university. The subject is often frowned upon and labeled as a "useless" subject, but I believe that any good mathematician, physicist, engineer, or architect should see an exposition of the subject to some degree. Not only is it an elementary foundation for whatever geometry one will need, but also it is a great source of beauty in math and great practice in creativity.
I will be assuming a very basic understanding of geometry, which includes results about angles, congruency and similarity of triangles and very basic geometry of the circle. (This presentation should be accessible even to middle school students with enough mathematical maturity.)
This presentation is concerned exactly with what I've said it's about, collinearity and concurrency, that is, when do three points lie on a line and when do three lines pass through a common point. These questions may sound simple and a bit mundane, but we will see many beautiful results, some of which discovered by the ancient greeks and some of which had elementary proofs discovered in the 20th century, that show collinearity and concurrency in some surprising configurations. Moreover, we will see a strange duality between the concepts of collinearity and concurrency (and therefore points and lines in a way).
We will start off with general results that give both \textit{necessary and sufficient} conditions for collinearity and concurrency in a given $\triangle ABC$. In fact, if points $D, E, F$ lie on $BC, CA, AB$ respectively, then the collinearity of $D, E, F$ and the concurrency of $AD, BE, CF$ depends only on the ratio $\frac{BD}{DC}\frac{CE}{EA}\frac{AF}{FB}$. We will quickly see some immediate corollaries of these powerful theorems. Then we will turn our attention to more special configurations and the related theorems, such as Euler's line, the Simson line, Pappus's theorem, and Desargues's Theorem to name a few. After developing some results on power of a point and radical axes, we will be able to prove the celebrated theorems of Pascal and Brianchon, which will naturally give rise to questions that belong in the realm of projective geometry, which we may touch on.
I hope this brief overview has peeked your interest and got you excited for this talk.
Mikey