Triangle geometry
The triangular geometry plays in the flat ( Euclidean ) geometry a special role because any polygons in triangles can be assembled. A clear differentiation from trigonometry , which is largely concerned with triangular calculations, is often not possible. Trigonometry is characterized by the use of the trigonometric functions ( sine , cosine , tangent , cotangent , secant , coscan ) and the emphasis on the computational aspect, while triangular geometry deals in general with properties of general and special triangles.
Triangle geometry is based on the sentences about sides and angles of the general triangle, some of which are dealt with in school geometry (for example about the sum of angles ) and the knowledge about special types of triangles:
- Isosceles triangle (equality of base angles)
- Equilateral triangle (60 ° angle)
- Right triangle ( Thales circle , Pythagorean theorem )
The "classical" transversals of the triangle were already examined in ancient Greek mathematics :
- the perpendicular lines (line symmetries ) that intersect at the circumcenter ,
- the bisectors (angular symmetries) of the inner and outer angles that intersect in the inscribed center point or in the circle centers,
- the side bisectors (median lines) that intersect (in a ratio of 2: 1) in the center of gravity ,
- and the heights that intersect in the orthocenter .
Only in the modern era (since the 17th century) were further discoveries made, including a large number of special points such as Fermat point , center point , nail point , Napoleon point , Lemoine point and Brocard point .
Play a particularly important role in triangle geometry
- the Euler's straight , on the circumcenter, centroid and orthocenter lie,
- and the Feuerbach circle (nine-point circle), which goes through the side centers, the height base points and the centers of the upper height sections and touches both the inscribed circle and the three approach circles.
Many discoveries of triangular geometry date back to the last two decades. The reason for this is not least the use of dynamic geometry software , which enables the creation of precise drawings with little expenditure of time and, in pull mode, quickly shows whether a guess could be generally correct or not. Computer programs for automated evidence are also used with success in this area. Another important tool with which the many special points of the triangle can be uniformly described are the trilinear and barycentric coordinates .
See also
literature
- Peter Baptist: The development of the newer triangular geometry (= textbooks and monographs on didactics of mathematics . Volume 19 ). BI-Wissenschaftsverlag, Mannheim u. a. 1992, ISBN 3-411-15661-9 (habilitation thesis, University of Würzburg, 1991).
- Philip J. Davis: The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-History . In: American Mathematical Monthly . tape 102 , no. 3 , March 1995, p. 204-214 , JSTOR : 2975007 .
- Wolfgang Grundmann: triangular geometry . Collection of special points, lines and circles on the triangle. 1st edition. AVM Akademische Verlagsgemeinschaft, Munich 2010, ISBN 978-3-89975-808-5 .