Circumconic and inconic

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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

[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

[edit] External links

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