The Feuerbach theorem in universal geometry

Abstract: 

Geometric proofs are often the subject of revisiting and streamlining as new perspectives and tools emerge in the world of mathematics. Tangential to this exercise is the ever-growing web that is the study of triangle centres, and at the point of tangency we can find the Feuerbach theorem and the Feuerbach point. This thesis will revisit this theorem once more, this time from the perspective of a universal geometry founded upon linear algebra, working without the assumptions that accompanied previous methods. Such an approach results in some nice expressions which leads to a deeper understanding of the Feuerbach theorem. And of course, because this is exploration is geometric in nature, we also stumble across some tangential discoveries concerning the Feuerbach point's relation to another triangle centre.