Circumconic and inconic
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 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
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
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
The lines tangent to the general circumconic at the vertices A,B,C are, respectively,
The center of the general inconic is the point
The lines tangent to the general inconic are the sidelines of ΔABC, given by the equations x = 0, y = 0, z = 0.
and to a rectangular hyperbola if and only if
As the parameter t ranges through the real numbers, the locus of X is a line. Define
The locus of X2 is the inconic, necessarily an ellipse
, given by the equation
where
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a circumconic is a conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
that passes through three given points, and an inconic is a conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
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
Trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous 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
Isogonal conjugate
In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P...
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 coordinatesTrilinear coordinatesIn geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous 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 parabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
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 parabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
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
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, 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.
- The lines connecting the tangency points of the inellipse of a triangle with the opposite vertices of the triangle are concurrent.
- Of all triangles inscribed in a given ellipse, the centroid of the one with greatest area coincides with the center of the ellipse.
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.Examples
- Circumconics
- circumcircle
- Steiner circumellipse
- Kiepert hyperbola
- Jeřábek hyperbola
- Feuerbach hyperbola
- Inconics
- incircle
- Steiner inellipseSteiner inellipseIn geometry, the Steiner inellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an inconic. By comparison the inscribed circle of a triangle is another inconic that is tangent to the sides, but not necessarily at the...
- Mandart inellipse
- Kiepert parabola
- Yff parabola
External links
- Circumconic at MathWorld
- Inconic at MathWorld