The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points.
Hence in trilinear co-ordinates, with ABC as fundamental triangle, its equation is Pa+Q/1+R7=o.
Trilinear and Tangential Co-ordinates.---The Geometrie descriptive, by Gaspard Monge, was written in the year 1794 or 1 795 (7th edition, Paris, 1847), and in it we have stated, in piano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, viz.
A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear co-ordinates.
The corresponding equations in areal co-ordinates are readily derived by substituting x/a, ylb, z/c for a, 1 3, y respectively in the trilinear equations.