In graph theory, I remember the first time proving $K_5$ is non-planar, and I thought it was pretty slick to use Euler's Theorem and some other results to prove this. However, in the back of my mind, I thought there could be another way of doing this that involved angles, vectors, or a constructive argument. There are a lot of results in geometry involving triangulation and other spatial theorems, and I would be curious if there was a way to do it this way.
Has anyone seen or able to prove a particular graph is non-planar by using angles, vectors, or a constructive argument?
We could use geometry to prove that $K_5$ has no straight-line embedding, for example.
To do this, first argue that three of the points $A,B,C$ must form a triangle containing the other two points $D,E$; there's several ways to do this. Then, the line $DE$ can only intersect two sides of $\triangle ABC$ (I mean the line segments); suppose it intersects $AB$ and $AC$. Then $BCDE$ is a convex quadrilateral, so its diagonals intersect.
Proving that $K_5$ is not planar is much harder. For the above argument to be enough, we would need to invoke Fáry's theorem that every planar graph has a straight-line embedding. This comes down to Euler's formula, among other things :)
Geometric arguments are used in some problems related to planar graphs, where this is not an issue. For example, Kurz and Pinchasi's proof that there is no $5$-regular matchstick graph is all about looking at the angles of various faces. (A matchstick graph is a graph with a planar embedding in which all the edges are unit line segments: we could represent it by putting matchsticks together.)