## Posts tagged ‘triangle’

### Casey’s Treatise on analytical geometry

John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized version of the book (see there).

The book begins with the definitions of classical coordinate systems: cartesian coordinates, polar coordinates, projective homogeneous coordinates (which were then dubbed trilinear or barycentric). Casey has no taboo concerning points of the field of complex numbers: in particular, many statements make use of cyclic points or zero-radius circles. Much further into the book, the author introduces projections and perspectives, as well as quadratic cones, and harmonic divisions.

### Un peu de géométrie plane…

Commençons par un peu de géométrie élémentaire… Soit $ABC$ un triangle non dégénéré dans le plan euclidien. L’usage de coordonnées barycentriques permet de repérer un point $M$ du plan par trois nombres réels ${[x:y:z]}$ de sorte que $M$ soit le barycentre des sommets du triangle avec les poids indiqués.

Let’s begin with elementary geometry… Let $ABC$ be a non-degenerate triangle in the Euclidean plane. By using barycentric coordinates, any point $M$ in the plan can be assigned a triple of real numbers ${[x:y:z]}$, such that $M$ is the barycentre of the vertices with these weights.