Circumconic and inconic

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In triangle geometry, a circumconic is a conic section that passes through the three vertices of a triangle, [1] and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle. [2]

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Suppose A,B,C are distinct non-collinear 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

Circumconic

The center of the general circumconic is the point

u(au + bv + cw) : v(aubv + cw) : w(au + bvcw).

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

wv + vz = 0,
uz + wx = 0,
vx + uy = 0.

Inconic

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.

Other features

Circumconic

(cxaz)(aybx) : (aybx)(bzcy) : (bzcy)(cxaz)
(vr + wq)x + (wp + ur)y + (uq + vp)z = 0.
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.

Inconic

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.
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 = q1r2r1q2,
M = r1p2p1r2,
N = p1q2q1p2.
which is maximized by the centroid's barycentric coordinates

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. [3] :p.136

Examples

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Steiner ellipse

In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse whose center is the triangle's centroid. Named after Jakob Steiner, it is an example of a circumconic. By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral.

Inellipse

In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, the Mandart inellipse and Brocard inellipse. For any triangle there exist an infinite number of inellipses.

References

  1. Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Circumconic.html
  2. Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html
  3. 1 2 3 4 5 6 7 Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.