Morley-Dreieck

Morley-Dreieck DEF (rot)

Das Morley-Dreieck, benannt nach Frank Morley, ist ein gleichseitiges Dreieck, welches innerhalb eines beliebigen Dreiecks konstruiert werden kann.

Definition und Satz von Morley

Die Innenwinkel eines beliebigen Dreiecks werden jeweils in drei gleich große Winkel unterteilt (was im Allgemeinen mit Zirkel und Lineal nicht möglich ist). Zu jeder Dreiecksseite betrachtet man den Schnittpunkt derjenigen zwei Teilungslinien, die von den Endpunkten dieser Seite ausgehen und zu dieser Seite benachbart sind. Das Morley-Dreieck ist dasjenige Dreieck, dessen Ecken die drei auf diese Weise erhaltenen Schnittpunkte sind.

Der Satz von Morley lautet:[1] Unabhängig von der Form des ursprünglichen Dreiecks ist das Morley-Dreieck stets gleichseitig.

Siehe auch

Ausgezeichnete Punkte im Dreieck

Literatur

  • H. S. M. Coxeter, S. L. Greitzer: Zeitlose Geometrie. Klett, Stuttgart 1983.
  • Horst Hischer: Grundlegende Begriffe der Mathematik: Entstehung und Entwicklung: Struktur – Funktion – Zahl. Springer 2012, ISBN 978-3-8348-8632-3, S. 2–4
  • Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, ISBN 978-3-662-53034-4, S. 182–185
  • Martin Erickson: Mathematische Appetithäppchen: Faszinierende Bilder. Packende Formeln. Reizvolle Sätze. Springer, 2015, ISBN 978-3-662-45459-6, S. 60–63 (mit Conways elegantem Beweis)

Einzelnachweise

  1. ↑ Max Koecher, Aloys Krieg: Ebene Geometrie. 3. Auflage. Springer-Verlag, Berlin 2007, ISBN 978-3-540-49327-3, S. 131

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