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In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Mandart inellipse 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 ...
is an ellipse inscribed within the triangle, tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to its 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 (which are also the vertices of the extouch triangle
In Euclidean geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle.
Coordinates
The vertex (geometry), vertices of the extouch triangle are given in trilinear coordinates by: ...
and the endpoints of the splitters). The Mandart inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century..;
. As cited by .
Parameters
As aninconic
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.Weisstein, Eric W. "Inconic." From MathWorld ...
, the Mandart inellipse is described by the parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
:
where ''a'', ''b'', and ''c'' are sides of the given triangle.
Related points
The center of the Mandart inellipse is themittenpunkt
In geometry, the (from German: ''middle point'') of a triangle is a triangle center: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It was identified in 1836 by Christian Heinrich von Nagel ...
of the triangle. The three lines connecting the triangle vertices to the opposite points of tangency all meet in a single point, the 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 concur ...
of the triangle.
See also
*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 insc ...
, a different ellipse tangent to a triangle at the midpoints of its sides
Notes
External links
*{{mathworld, urlname=MandartInellipse, title=Mandart Inellipse Curves defined for a triangle