Bisector set (triangle)

${\displaystyle \gamma _{1}=\gamma _{2}\,\Leftrightarrow \,{\tfrac {|AC|}{|BC|}}={\tfrac {|AD|}{|BD|}}}$

The angle bisector theorem is a statement of the elementary geometry. It says that the bisector in a triangle divides the side opposite the angle in relation to the two sides adjacent to the angle.

Theorem and generalization

In a triangle, let a point be on the side . The segment divides the angle into angles and . If these two angles are the same size, i.e. the bisector of the angle , then the following applies to the route ratios: ${\displaystyle \triangle ABC}$ ${\displaystyle D}$ ${\displaystyle AB}$ ${\displaystyle CD}$ ${\displaystyle \angle ACB}$ ${\displaystyle \gamma _{1}=\angle ACD}$ ${\displaystyle \gamma _{2}=\angle DCB}$ ${\displaystyle CD}$ ${\displaystyle \angle ACB}$

${\displaystyle {\frac {|AC|}{|BC|}}={\frac {|AD|}{|DB|}}}$.

This statement can also be generalized to segments that divide the angle in any ratio. The following equation then applies: ${\displaystyle CD}$

${\displaystyle {\frac {|AC|\sin(\gamma _{1})}{|BC|\sin(\gamma _{2})}}={\frac {|AD|}{|DB|}}}$.

The reverse of the angle bisector also applies. That is, if there is a point on the side of a triangle and the distance ratio applies , then the bisector of the angle is in . ${\displaystyle D}$ ${\displaystyle AB}$ ${\displaystyle \triangle ABC}$ ${\displaystyle {\tfrac {|AC|}{|BC|}}={\tfrac {|AD|}{|DB|}}}$ ${\displaystyle CD}$ ${\displaystyle C}$

proof

Sketch for proof

A simple proof of the generalized statement can be obtained by taking the quotient of the areas of the two partial triangles created by the bisector and calculating this in two different ways. The first time is used to calculate the triangular faces, the formula with the base side and the associated height and the second time, the formula with the two sides , and of an included angle . ${\displaystyle {\tfrac {1}{2}}gh}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle {\tfrac {1}{2}}ab\sin(\gamma )}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \gamma }$

So you get now

${\displaystyle {\frac {|\triangle ADC|}{|\triangle DBC|}}={\frac {{\frac {1}{2}}|AD|h}{{\frac {1}{2}}|BD|h}}={\frac {|AD|}{|BD|}}}$

and

${\displaystyle {\frac {|\triangle ADC|}{|\triangle BDC|}}={\frac {{\frac {1}{2}}|AC||AD|\sin(\gamma _{1})}{{\frac {1}{2}}|BC||AD|\sin(\gamma _{2})}}={\frac {|AC|\sin(\gamma _{1})}{|BC|\sin(\gamma _{2})}}}$

therefore applies

${\displaystyle {\frac {|AC|\sin(\gamma _{1})}{|BC|\sin(\gamma _{2})}}={\frac {|AD|}{|BD|}}}$

Outside angle bisector

Außenwinkelhablierenden (red dashed line):
The three intersection points D, E, F lie on a line (red) and apply the following track conditions: , ,
${\displaystyle {\tfrac {|EB|}{|EC|}}={\tfrac {|AB|}{|AC|}}}$ ${\displaystyle {\tfrac {|FB|}{|FA|}}={\tfrac {|CB|}{|CA|}}}$ ${\displaystyle {\tfrac {|DA|}{|DC|}}={\tfrac {|BA|}{|BC|}}}$

If it is not an equilateral triangle, there are also ratio equations for the outer angle bisectors of a triangle, which contain the sides of the triangle. More specifically, for a non-equilateral triangle, the following applies . Cuts the Außenwinkelhalbierende in the extension of the side in which Außenwinkelhalbierende in the extension of the side in and the Außenwinkelhalbierende in the extension of the side in , then: ${\displaystyle ABC}$ ${\displaystyle A}$ ${\displaystyle BC}$ ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle AC}$ ${\displaystyle D}$ ${\displaystyle C}$ ${\displaystyle AB}$ ${\displaystyle F}$

${\displaystyle {\frac {|EB|}{|EC|}}={\frac {|AB|}{|AC|}}}$, and ${\displaystyle {\frac {|FB|}{|FA|}}={\frac {|CB|}{|CA|}}}$ ${\displaystyle {\frac {|DA|}{|DC|}}={\frac {|BA|}{|BC|}}}$

In addition, the points , and lie on a common straight line. ${\displaystyle D}$ ${\displaystyle E}$ ${\displaystyle F}$

history

The Winkelhalbierendensatz can already be found in Euclid in the elements in the book VI as Proposition third

literature

• Siegfried Krauter, Christine Bescherer: The Elementary Geometry Experience: A workbook for independent and active discovery . Springer, 2012, ISBN 978-3-8274-3025-0 , p. 161
• Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer, 2015, ISBN 978-3-662-45461-9 , p. 66

Web links

• proof of the angle bisector theorem (video) at the Khan Academy (English)
• angle bisector theorem on cut-the-knot.org

Individual evidence

1. ^ Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer, 2015, ISBN 978-3-662-45461-9 , p. 66
2. ↑ Titu Andreescu, Zuming Feng: 103 Trigonometry problem: From the Training of the USA IMO Team . Springer, 2006, p. 19
3. ^ Alfred S. Posamentier: Advanced Euclidian Geometry: Excursions for Students and Teachers . Springer, 2002, ISBN 9781930190856 , pp. 3-4
4. ^ Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , p. 149 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).
5. ^ Isaac Todhunter , The Elements of Euclid , Book VI, Proposition 3.