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.