Base triangle

Triangle ABC (red), perpendicular from point  P to the sides (green), base triangle from P (blue)

Base triangle is a term from triangle geometry . If a triangle ABC and a point P are given, the base point triangle of P is given by the base points of the three perpendiculars from P on the sides of the triangle. If P lies on the periphery of ABC, the base point triangle degenerates into a straight line, the so-called Samson straight line .

If the given point P is the vertical intersection of the triangle, one speaks of the height base triangle .

The side lengths of a base triangle can be calculated from the side lengths of the original triangle, the distances from its corner points to point P and the radius r of the circumference. The following applies:

${\displaystyle {\begin{aligned}&|MN|&={\frac {|AP|\cdot |BC|}{2r}}\\&|NL|&={\frac {|BP|\cdot |AC|}{2r}}\\&|LM|&={\frac {|CP|\cdot |AB|}{2r}}\\\end{aligned}}}$

These relationships also apply in the case of the degenerate triangle, if P lies on the circumference and the corresponding route sections on the Samson straight line.

literature

• HSM Coxeter , SL Greitzer: Timeless geometry . Klett, Stuttgart 1983
• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 121–127 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry , pp. 135–139)
• J. Vályi: About base triangles . In: Monthly books for mathematics , Volume 14, No. 1, December 1903, Springer

Web links

• Eric W. Weisstein : Pedal Triangle . In: MathWorld (English).
• Height base triangle - a visualization with GeoGebra