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Concept
Circumconic and inconic
Summary
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.
Suppose A, B, C are distinct non-collinear points, and let △ABC 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 △ABC. Let the sidelengths of △ABC.
In trilinear coordinates, the general circumconic is the locus of a variable point 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 △ABC 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 △ABC and is given by the equation
The center of the general circumconic is the point
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 △ABC, given by the equations x = 0, y = 0, z = 0.
Each noncircular circumconic meets the circumcircle of △ABC in a point other than A, B, C, often called the fourth point of intersection, given by trilinear coordinates
If is a point on the general circumconic, then the line tangent to the conic at P is given by
The general circumconic reduces to a parabola if and only if
and to a rectangular hyperbola if and only if
Of all triangles inscribed in a given ellipse, the centroid 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.
The general inconic reduces to a parabola if and only if
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.
Official source
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