Circumconic and inconic

From Wikipedia, the free encyclopedia

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 of three given points.

Suppose A,B,C are distinct point, 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.

Contents 1 Centers and tangent lines 2 Other features 3 Examples 4 External Links

[edit] 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.

[edit] 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.

[edit] Examples

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

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

[edit] External Links

• Circumconic at MathWorld
• Inconic at MathWorld