Circumconic and inconic
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle. Suppose A, B, C are distinct non-collinear points, and let △ABC denote the triangle whose vertices are A, B, C. Following common practice, A denotes not only the vertex but also the angle ∠BAC at vertex A, and similarly for B and C as angles in △ABC. Let a = | B C | , b = | C A | , c = | A B | , {\displaystyle a=|BC|,b=|CA|,c=|AB|,} the sidelengths of △ABC. In trilinear coordinates, the general circumconic is the locus of a variable point X = x : y : z {\displaystyle X=x:y:z} satisfying an equation u y z + v z x + w x y = 0 , {\displaystyle uyz+vzx+wxy=0,} for some point u : v : w.