Circumconic and inconic

The center of the general circumconic is the point

u(−au + bv + cw) : v(au − bv + cw) : w(au + bv − cw).

The lines tangent to the general circumconic at the vertices A,B,C are, respectively,

wv + vz = 0,

uz + wx = 0,

vx + uy = 0.

The center of the general inconic is the point

cv + bw : aw + cu : bu + av.

The lines tangent to the general inconic are the sidelines of ΔABC, given by the equations x = 0, y = 0, z = 0.

• 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

(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 P is given by

(vr + wq)x + (wp + ur)y + (uq + vp)z = 0.

• The general circumconic reduces to a parabola if and only if

u²a² + v²b² + w²c² − 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.[3]^:p.147 The given ellipse, going through this triangle's three vertices and centered at the triangle's centroid, is called the triangle's Steiner circumellipse.

• The general inconic reduces to a parabola if and only if

ubc + vca + wab = 0,

in which case it is tangent externally to one of the sides of the triangle and is tangent to the extensions of the other two sides.

• Suppose that p₁ : q₁ : r₁ and p₂ : q₂ : r₂ are distinct points, and let

X = (p₁ + p₂t) : (q₁ + q₂t) : (r₁ + r₂t).

As the parameter t ranges through the real numbers, the locus of X is a line. Define

X² = (p₁ + p₂t)² : (q₁ + q₂t)² : (r₁ + r₂t)².

The locus of X² is the inconic, necessarily an ellipse, given by the equation

L⁴x² + M⁴y² + N⁴z² − 2M²N²yz − 2N²L²zx − 2L²M²xy = 0,

where

L = q₁r₂ − r₁q₂,

M = r₁p₂ − p₁r₂,

N = p₁q₂ − q₁p₂.

• 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.[3]^:p.139 For a given point inside that medial triangle, the inellipse with its center at that point is unique.[3]^:p.142
• The inellipse with the largest area is the Steiner inellipse, also called the midpoint inellipse, with its center at the triangle's centroid.[3]^:p.145 In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum barycentric coordinates ${\displaystyle (\alpha ,\beta ,\gamma )}$ of the inellipse's center, is[3]^:p.143

${\displaystyle {\frac {\text{Area of inellipse}}{\text{Area of triangle}}}=\pi {\sqrt {(1-2\alpha )(1-2\beta )(1-2\gamma )}},}$

which is maximized by the centroid's barycentric coordinates ${\displaystyle \alpha =\beta =\gamma =1/3.}$

• The lines connecting the tangency points of any inellipse of a triangle with the opposite vertices of the triangle are concurrent.[3]^:p.148