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I am presently teaching eleventh grade (XI standard) students an introductory course in co-ordinate geometry with a focus on preparations for competitive exams. I have seen books like S.L.Loney's co-ordinate geometry which is like an encyclopedia of co-ordinate geometric results. I feel that an encyclopedia does not actually give a good taste of a field. So I wanted interesting results and techniques that arise from co-ordinate geometry.

My present focus is points and straight lines.

I have observed that co-ordinate geometry has very mysteriously simple formulas for many problems. For example, if $L(x,y) = ax+by-c$ where $a^2+b^2 =1$, if $P$ is any point on the plane, $L(P)$ represents the perpendicular distance of the point from the plane (even the sign has meaning). It follows that if two such lines $L_1$ and $L_2$ are given, then $L_1 \pm L_2 = 0$ represent the two angle bisectors through the intersection point of $L_1$ and $L_2$.

I thought that this behavior is due to vectors and started teaching the course by viewing a line $L$ as $(a,b)\cdot(x,y)^T = c \implies \overline{w}\cdot \overline{r} = c.$ Now associated with every line $L: \overline{w}\cdot \overline{r} = c$ not passing through origin, we have a unique vector $\frac{\overline{w}}{c}$. From now on we will assume $c=1$. A cool theorem here is:

A family of lines $L(\overline{w})$ is concurrent at a point iff the associated vectors $\overline{w}$ are collinear.

I was enthralled by this observation. However this quickly started to fade when I realized I could not get a meaningful translation to all the formulae of the triangle. I am looking at other approaches like Mobius's barycentric co-ordinates and projective geometric notions (by going to three dimensions and then doing two dimensional geometry on a sliced plane).

So I have the following precise questions:

1) Are there simple formulas for the medians, altitudes and other cevians in terms of $\overline{w}$ if one is given three lines $L(\overline{w_1}),L(\overline{w_2}),L(\overline{w_3})$ which form the sides of the triangle?

2) Are there alternative approaches to points and straight lines which makes certain proofs (and concepts) simpler?

Note: I am aware of the duality between lines and points used in modern expositions of vector spaces (functional analysis) and I know that is related to my exposition. I am also aware of Poncelot duality and the projective geometric idea of duality in theorems about collineations (Pascal's theorem, Desargues theorem and so on). I am aware of Hilbert spaces which are complete vector spaces endowed with a notion of angle and Banach spaces are complete spaces endowed with a distance. However, I want to connect these modern viewpoints to concrete non-trivial elementary examples. So I would be happy if a knowledgeable reader shares examples of theorems in co-rdinate geometry which have deeper analogs in advanced mathematics.

Thanks!

P.S: I asked this question on math stackexchange today morning and I was directed to this site. See the question here: https://math.stackexchange.com/questions/1454723/on-teaching-elementary-co-ordinate-geometry

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    $\begingroup$ I challenge the assertion that concurrency occurs when the w's are collinear. If I understand things correctly, w's being collinear mean they point in the same direction, and thus the associated lines are parallel, as w is a normal to its line. Concurrency relates to solving a (possibly overdetermined) system of linear equations, but I do not see a nice interpretation based on L(w). Gerhard "Don't Know About The Formulas" Paseman, 2015.09.28 $\endgroup$ Sep 28, 2015 at 17:07
  • $\begingroup$ Hi Gerhard. The lines are parallel if the $w$ vectors are collinear with $0$. If the $w$ vectors are collinear and $0$ does not lie on this line, then the associated lines are concurrent. Your claim follows from observing that the vector $w=0$ represents the line at infinity. The line at infinity is made of all points that are infinite, one for each direction. $\endgroup$
    – Spai
    Sep 28, 2015 at 23:04
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    $\begingroup$ Sorry, but my eyes bleed each time I read "co-ordinates". Is this really an accepted spelling (I am French, so please pardon my ignorance)? $\endgroup$ Sep 29, 2015 at 7:31
  • $\begingroup$ @BenoîtKloeckner: I think both coordinates and co-ordinates are accepted spellings. See thefreedictionary.com/co-ordinate $\endgroup$
    – Spai
    Sep 30, 2015 at 8:23
  • $\begingroup$ @Spai Wow! I came here from reading your posts on MSE, the IIT JEE link you had put in comments of one of your questions caught my eye. The questions you ask and the way you think of these things are very inspirational to me! $\endgroup$ Feb 4, 2021 at 9:32

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