Elevation base triangle

Elevation base triangle

The height base triangle (more rarely: orthic triangle ) is a term from triangular geometry . It arises from the fact that the base points of the three heights (i.e. the points , and , at which the perpendiculars from the corners of the triangle to the opposite sides intersect these sides) are connected to one another. In the special case of a right triangle, the height base triangle is degenerate, since two base points then coincide. The “height base triangle” is the base triangle belonging to the height intersection (orthocenter) . ${\displaystyle H_{a}}$ ${\displaystyle H_{b}}$ ${\displaystyle H_{c}}$

Similar height base triangle to the tangent triangle

properties

• Each height of the original triangle bisects either an interior angle or an exterior angle of the height base triangle. Therefore, for an acute-angled triangle ABC, the height intersection point  H of this triangle coincides with the inscribed center of the height base point triangle. On the other hand , if the triangle ABC is obtuse-angled , then H is equal to one of the circle centers of the base point triangle.
• The perimeter of the height base triangle is the Feuerbach circle of the original triangle.
• Fagnano problem : Of all the triangles inscribed in an acute-angled triangle, the height base triangle has the smallest circumference .

See also

Excellent points in the triangle

literature

• Harold SM Coxeter , Samuel L. Greitzer: Timeless Geometry. Klett, Stuttgart 1983, ISBN 3-12-983390-0 .

Web links

• Elevation points - a visualization with GeoGebra
• Eric W. Weisstein : Orthic Triangle . In: MathWorld (English).
• Orthic Triangle (PDF; 104 kB)

Individual evidence

1. ↑ Max Koecher , Aloys Krieg : level geometry. 3rd, revised and expanded edition. Springer, Berlin a. a. 2007, ISBN 978-3-540-49327-3 , p. 168.