Circumconic and inconic

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]

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


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


(cx az)(ay bx) : (ay bx)(bz cy) : (bz cy)(cx az)
(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.


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,
L = q1r2 r1q2,
M = r1p2 p1r2,
N = p1q2 q1p2.
\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 \alpha =\beta = \gamma = 1/3.

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



  1. Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web Resource.
  2. Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource.
  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.

External links

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