Circumconic and inconic
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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, a circumconic is a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
that passes through the three vertices of a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
, and an inconic is a conic section
inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html Suppose are distinct non-collinear points, and let denote the triangle whose vertices are . Following common practice, denotes not only the vertex but also the angle at vertex , and similarly for and as angles in . Let a= , BC, , b=, CA, , c=, AB, , the sidelengths of . In
trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
, the general circumconic is the locus of a variable point X = x:y:z satisfying an equation :uyz + vzx + wxy = 0, for some point . The
isogonal conjugate __notoc__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...
of each point on the circumconic, other than , is a point on the line :ux + vy + wz = 0. This line meets the circumcircle of 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 and is given by the equation :u^2x^2 + v^2y^2 + w^2z^2 - 2vwyz - 2wuzx - 2uvxy = 0.


Centers and tangent lines


Circumconic

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 are, respectively, :\begin wv+vz &= 0, \\ uz+wx &= 0, \\ vx+uy &= 0. \end


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 , given by the equations , , .


Other features


Circumconic

* Each noncircular circumconic meets the circumcircle of in a point other than , often called the fourth point of intersection, given by
trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
:: (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 is given by :: (vr+wq)x + (wp+ur)y + (uq+vp)z = 0. * The general circumconic reduces to a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
if and only if :: u^2a^2 + v^2b^2 + w^2c^2 - 2vwbc - 2wuca - 2uvab = 0, :and to a
rectangular hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
if and only if :: u\cos A + v\cos B + w\cos C = 0. * Of all triangles inscribed in a given ellipse, the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
of the one with greatest area coincides with the center of the ellipse. The given ellipse, going through this triangle's three vertices and centered at the triangle's centroid, is called the triangle's
Steiner circumellipse In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse ( ellipse that touches the triangle at its vertices) whose center is the triangle's ...
.


Inconic

* The general inconic reduces to a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
if and only if :: 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. * Suppose that and are distinct points, and let ::X = (p_1+p_2 t) : (q_1+q_2 t) : (r_1+r_2 t). :As the parameter ranges through the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, the locus of is a line. Define :: X^2 = (p_1+p_2 t)^2 : (q_1+q_2 t)^2 : (r_1+r_2 t)^2. :The locus of is the inconic, necessarily an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, given by the equation ::L^4x^2 + M^4y^2 + N^4z^2 - 2M^2N^2yz - 2N^2L^2zx - 2L^2M^2xy = 0, :where ::\begin L &= q_1r_2 - r_1q_2, \\ M &= r_1p_2 - p_1r_2, \\ N &= p_1q_2 - q_1p_2. \end * 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.Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. For a given point inside that
medial triangle In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is n ...
, the inellipse with its center at that point is unique. * The inellipse with the largest area is the
Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse i ...
, also called the midpoint inellipse, with its center at the triangle's
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
. In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum
barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
of the inellipse's center, is ::\frac= \pi \sqrt, :which is maximized by the centroid's barycentric coordinates . * The lines connecting the tangency points of any inellipse of a triangle with the opposite vertices of the triangle are concurrent.


Extension to quadrilaterals

All the centers of inellipses of a given
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
fall on the line segment connecting the midpoints of the
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s of the quadrilateral.


Examples

* Circumconics **
Circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
, the unique
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
that passes through a triangle's three vertices **
Steiner circumellipse In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse ( ellipse that touches the triangle at its vertices) whose center is the triangle's ...
, the unique ellipse that passes through a triangle's three vertices and is centered at the triangle's
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
**
Kiepert hyperbola *Friedrich Wilhelm August Ludwig Kiepert Friedrich Wilhelm August Ludwig Kiepert (6 October 1846 – 5 September 1934) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typic ...
, the unique conic which passes through a triangle's three vertices, its centroid, and its
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
** Jeřábek hyperbola, a
rectangular hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
centered on a triangle's
nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of ea ...
and passing through the triangle's three vertices as well as its
circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
, orthocenter, and various other notable centers **
Feuerbach hyperbola In geometry, the Feuerbach hyperbola is a rectangular hyperbola passing through important triangle centers such as the Orthocenter, Gergonne Point, Gergonne point, Nagel point and Schiffler point, Shiffler point. The center of the hyperbola is the ...
, a rectangular hyperbola that passes through a triangle's orthocenter,
Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concu ...
, and various other notable points, and has center on the nine-point circle. * Inconics **
Incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
, the unique circle that is internally tangent to a triangle's three sides **
Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse i ...
, the unique ellipse that is tangent to a triangle's three sides at their midpoints **
Mandart inellipse In geometry, the Mandart inellipse of a triangle is an ellipse inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle and the endpoints of the splitters). ...
, the unique ellipse tangent to a triangle's sides at the contact points of its
excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
s **
Kiepert parabola In Modern triangle geometry, triangle geometry, the Kiepert conics are two special conics associated with the reference triangle. One of them is a hyperbola, called the Kiepert hyperbola and the other is a parabola, called the Kiepert parabola. The ...
** Yff parabola


References

{{reflist


External links


Circumconic
at MathWorld

at MathWorld Conic sections