Circumconic • Each noncircular circumconic meets the circumcircle of in a point other than , often called the
fourth point of intersection, given by
trilinear coordinates :: (cx-az)(ay-bx) : (ay-bx)(bz-cy) : (bz-cy)(cx-az) • If P = p:q:r is a point on the general circumconic, then the line tangent to the conic at is given by :: (vr+wq)x + (wp+ur)y + (uq+vp)z = 0. • The general circumconic reduces to a
parabola if and only if :: u^2a^2 + v^2b^2 + w^2c^2 - 2vwbc - 2wuca - 2uvab = 0, :and to a
rectangular hyperbola if and only if :: u\cos A + v\cos B + w\cos C = 0. • Of all triangles inscribed in a given ellipse, the
centroid of the one with greatest area coincides with the center of the ellipse. For a given point inside that
medial triangle, the inellipse with its center at that point is unique. • The inellipse with the largest area is the
Steiner inellipse, also called the midpoint inellipse, with its center at the triangle's
centroid. In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum
barycentric coordinates of the inellipse's center, is ::\frac{\text{Area of inellipse}}{\text{Area of triangle}}= \pi \sqrt{(1-2\alpha)(1-2\beta)(1-2\gamma)}, :which is maximized by the centroid's barycentric coordinates . • The lines connecting the tangency points of any inellipse of a triangle with the opposite vertices of the triangle are concurrent. ==Extension to quadrilaterals==