Circumconic and inconic

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.

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Steiner inellipse

In geometry, the Steiner inellipse, midpoint inellipse, or midpoint ellipse 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 inellipse. By comparison the inscribed circle and Mandart inellipse of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is equilateral. The Steiner inellipse is attributed by Dörrie to Jakob Steiner, and a proof of its uniqueness is given by Dan Kalman.

Trilinear coordinates

In geometry, the trilinear coordinates x : y : z 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 x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z : x and vertices C and A.

Extended side

In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a finite side into an infinite line arises in various contexts. In an obtuse triangle, the altitudes from the acute angled vertices intersect the corresponding extended base sides but not the base sides themselves. The excircles of a triangle, as well as the triangle's inconics that are not inellipses, are externally tangent to one side and to the other two extended sides.

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