South pole set

The South Pole Theorem is a geometric theorem from triangle theory . It says that in a triangle the mid-line (line symmetry) on one side and the bisector (angle symmetry) through the opposite corner always intersect on the circumference . The intersection is also called the South Pole .

proof

The sentence should be justified for the vertical line and the bisector . S is defined as the intersection of the perpendicular with the circumference that is not on the same side of AB as corner C. If U is the circumcentric center, then the angle ACS is half as large as the angle AUS, since it is the circumferential angle (Peripheral angle ) and central angle (central angle) over the circular arc AS. Correspondingly, the angle SCB is half as large as the angle SUB. Since the angles AUS and SUB are the same size for reasons of symmetry , the angles ACS and SCB must also be the same. In other words: S must lie on the bisector of the angle . The intersection of the perpendicular and the bisector must therefore coincide with S and consequently lie on the circumference. ${\ displaystyle m_ {c}}$ ${\ displaystyle w _ {\ gamma}}$ ${\ displaystyle m_ {c}}$ ${\ displaystyle \ gamma = \ angle ACB}$

Since S lies on the perpendicular perpendicular to AB, the distances AS and BS must be of equal length for reasons of symmetry. According to the set of circumferential angles, the angles must be the same over a chord of the same length, so the angles in C over the chords AS and BS are also the same. And therefore CS is the bisector in C, which intersects with the center perpendicular on the circumference.

literature

HK Dass, Rama Verma, Bhagwat S. Sharma: S.Chand'S Mathematics For Class IX Term II . S. Chand Publishing, 2011, ISBN 9788121938464 , pp. 165-166

Web links

Franz Hofbauer: Geometry and linear algebra for teaching . Script, University of Vienna, page 24, sentence 33 (contains similar evidence)
Lucas Mann: Script for the MSG circle . Script of the mathematical student society Leonard Euler, page 51, exercise 5.16
Dörte Haftendorn : South Pole Theorem - Drawing and Proof (pdf)
Walter Fendt: The South Pole Sentence (interactive illustration)