Triangle Conic
In triangle geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steiner ellipse which is an ellipse passing through the vertices and having its centre at the centroid of the reference triangle, the Kiepert hyperbola which is a conic passing through the vertices, the centroid and the orthocentre of the reference triangle and the Artzt parabolas which are parabolas touching two sidelines of the reference triangle at vertices of the triangle. The terminology of ''triangle conic'' is widely used in the literature without a formal definition,that is, without precisely formulating the relations a conic should have with the reference triangle so as to qualify it to be called a triangle conic (see,). WolframMathWorld has a page titled "Triangle conics" which gives a list of 42 items (not all of them are conics) wit ...
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Triangle Geometry
A triangle is a polygon with three edges and three 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, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of Euclid's Elements. The names used for modern classification are eith ...
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Kiepert Hyperbola
*Friedrich Wilhelm August Ludwig Kiepert, German mathematician *Heinrich Kiepert, German Geographer *Richard Kiepert Richard Kiepert (September 13, 1846 – August 4, 1915) was a German cartographer born in Weimar. He was the son of famed geographer Heinrich Kiepert. Biography Richard Kiepert studied geography and history in Berlin and Heidelberg. In 1870, he ...

Kiepert Conics
In 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 Kiepert conics are defined as follows: :If the three triangles A^\prime BC, AB^\prime C and ABC^\prime, constructed on the sides of a triangle ABC as bases, are similar, isosceles and similarly situated, then the triangles ABC and A^\prime B^\prime C^\prime are in perspective. As the base angle of the isosceles triangles varies between -\pi/2 and \pi/2, the locus of the center of perspectivity of the triangles ABC and A^\prime B^\prime C^\prime is a hyperbola called the Kiepert hyperbola and the envelope of their axis of perspectivity is a parabola called the Kiepert parabola. It has been proved that the Kiepert hyperbola is the hyperbola passing through the vertices, the centroid and the orthocenter of the reference triangle and the Kiepert parabola is the pa ...
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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 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.. The Steiner inellipse contrasts with the Steiner circumellipse, also called simply the Steiner ellipse, which is the unique ellipse that passes through the vertices of a given triangle and whose center is the triangle's centroid. Definition and properties ;Definition An ellipse that is tangent to the sides of a triangle at its midpoints M_1 ...
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Lemoine Point
In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle. Ross Honsberger called its existence "one of the crown jewels of modern geometry". In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6).Encyclopedia of Triangle Centers
accessed 2014-11-06. For a non-equilateral triangle, it lies in the open punctured at its own center, and could be any point therein. The symmedian point of a triangle with side lengths , and has homogeneous

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, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ...
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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 ''extended base'' of the altitude. The intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometri ...
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