Arbitrary triangle △ABC
Extouch triangle△TATBTC
In Euclidean geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle.
In Euclidean geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle.
The vertices of the extouch triangle are given in trilinear coordinates by:
or equivalently, where a, b, c are the lengths of the sides opposite angles A, B, C respectively,
The triangle's splitters are lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet at the Nagel point. This is shown in blue and labelled "N" in the diagram.
The Mandart inellipse is tangent to the sides of the reference triangle at the three vertices of the extouch triangle. [1]
The area of the extouch triangle, KT, is given by:
where K and r are the area and radius of the incircle, s is the semiperimeter of the original triangle, and a, b, c are the side lengths of the original triangle.
This is the same area as that of the intouch triangle. [2]
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, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .
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 .
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In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
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In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.
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.
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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. The Mandart inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century.
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.
In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse whose center is the triangle's centroid. Named after Jakob Steiner, it is an example of a circumconic. By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral.
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.
In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, the Mandart inellipse and Brocard inellipse. For any triangle there exist an infinite number of inellipses.
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 without giving a definition of triangle conic. However, Paris Pamfilos in his extensive collection of topics in geometry and topics in other fields related to geometry defines a triangle conic as a "conic circumscribing a triangle ABC or inscribed in a triangle ". The terminology triangle circle is used to denote a circle associated with the reference triangle is some way.
