In triangle geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conicsectioninscribed in the sides, possibly extended, of a triangle. Suppose A,B,C are distinct non-collinear points, and letΔABCdenote 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 ΔABC. Let a = |BC|, b = |CA|, c = |AB|, the sidelengths of ΔABC. In trilinear coordinates, the general circumconic is the locus of a variable point X = x : y : z satisfying an equation for some point u : v : w. The isogonal conjugate of each point X on the circumconic, other than A,B,C, is a point on the line This line meets the circumcircle of ΔABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola. The general inconic is tangent to the three sidelines of ΔABC and is given by the equation
The center of the general circumconic is the point The lines tangent to the general circumconic at the vertices A,B,C are, respectively,
Inconic
The center of the general inconic is the point The lines tangent to the general inconic are the sidelines of ΔABC, given by the equations x = 0, y = 0, z = 0.
Other features
Circumconic
Each noncircular circumconic meets the circumcircle of ΔABC in a point other than A, B, and C, often called the fourth point of intersection, given by trilinear coordinates
If P = p : q : r is a point on the general circumconic, then the line tangent to the conic at P is given by
The general circumconic reduces to a parabola if and only if
Of all triangles inscribed in a given ellipse, the centroid of the one with greatest area coincides with the center of the ellipse. The given ellipse, going through this triangle's three vertices and centered at the triangle's centroid, is called the triangle's Steiner circumellipse.
Inconic
The general inconic reduces to a parabola if and only if
Suppose that p1 : q1 : r1 and p2 : q2 : r2 are distinct points, and let
A point in the interior of a triangle is the center of an inellipse of the triangle if and only if the point lies in the interior of the triangle whose vertices lie at the midpoints of the original triangle's sides. For a given point inside that medial triangle, the inellipse with its center at that point is unique.
* Feuerbach hyperbola, a rectangular hyperbola that passes through a triangle's orthocenter, Nagel point, and various other notable points, and has center on the nine-point circle.
Inconics
* Incircle, the unique circle that is internally tangent to a triangle's three sides
* Steiner inellipse, the unique ellipse that is tangent to a triangle's three sides at their midpoints
* Mandart inellipse, the unique ellipse tangent to a triangle's sides at the contact points of its excircles
