The Mandara ellipse (red) is inscribed in a triangle (black) at the points of contact of the sides with the extra circles (gray). Lines passing through the Nagel point - green; lines passing through the center of the ellipse - blue.
Mandara ellipse - an ellipse inscribed in a given triangle, touching its sides at the points of contact with non-inscribed circles [1] .
It is named after the French mathematician Mandar ( H. Mandart ), who published studies of this object in 1893-1894 [2] [3] .
The center of the Mandar ellipse is one of the remarkable points of a triangle ( German: mittenpunkt ), found by Nagel in 1836 as the intersection point of the triangles of the triangle formed by the centers of its incircle circles [4] [5] . In the Encyclopedia of Triangle Centers, an identifier is assigned to a point .
For inscribed conics, the inscribed Mandar ellipse is described by the parameters :
- ,
Where , and - sides of a given triangle.
Notes
- ↑ Juhász Imre. Control point based representation of inellipses of triangles // Annales Mathematicae et Informaticae. - 2012 .-- T. 40 . - S. 37–46 .
- ↑ Gibert, Bernard (2004), " Generalized Mandart conics ", Forum Geometricorum T. 4: 177–198 , < http://forumgeom.fau.edu/FG2004volume4/FG200421.pdf > .
- ↑ Mandart, H. (1893), "Sur l'hyperbole de Feuerbach", Mathesis : 81–89 ; Mandart, H. (1894), " Sur une ellipse associée au triangle ", Mathesis : 241–245 , < https://books.google.com/books?id=kqAKAAAAYAAJ&pg=PA241 > . As cited by Gibert (2004 )
- ↑ Kimberling, Clark (1994), " Central Points and Central Lines in the Plane of a Triangle ", Mathematics Magazine T. 67 (3): 163–187 , DOI 10.2307 / 2690608
- ↑ von Nagel, CH (1836), Untersuchungen über die wichtigsten zum Dreiecke gehörenden Kreise , Leipzig