Circumconic and inconic

In triangle geometry, a circumconic is a conic section that passes through three given points, and an inconic is a conic section inscribed in the triangle whose vertices lie at three given points.

Suppose A,B,C are distinct points, and let ΔABC 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 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

uyz + vzx + wxy = 0,

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

ux + vy + wz = 0.

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

u²x² + v²y² + w²z² − 2vwyz − 2wuzx − 2uvxy = 0.

Centers and tangent lines

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

cy + bz : az + cx : bx + ay.

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

Other features

• 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

x cos A + y cos B + z cos C = 0.

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

ubc + vca + wab = 0.

• 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.^[1]:p.139

• The lines connecting the tangency points of the inellipse of a triangle with the opposite vertices of the triangle are concurrent.^[1]:p.148

• Of all triangles inscribed in a given ellipse, the centroid of the one with greatest area coincides with the center of the ellipse.^[1]:p.147

Extension to quadrilaterals

All the centers of inellipses of a given quadrilateral fall on the line segment connecting the midpoints of the diagonals of the quadrilateral.^[1]:p.136

Examples

• Circumconics
  • circumcircle
  • Steiner circumellipse
  • Kiepert hyperbola
  • Jeřábek hyperbola
  • Feuerbach hyperbola

• Inconics
  • incircle
  • Steiner inellipse
  • Mandart inellipse
  • Kiepert parabola
  • Yff parabola

References

1. ^ ^{a b c d} Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.

External links

• Circumconic at MathWorld
• Inconic at MathWorld