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 would be happy if a knowledgeable reader shares examples of theorems in co-rdinate geometry which have deeper analogs in advanced mathematics.
Thanks!