Bisector

The bisectors in the triangle.
S, the intersection of the three bisectors, is the center of gravity of the triangle. He divides the two sides in a ratio of 2: 1.

A Seitenhalbierende (also heavy line or median ) in a triangle is a route that a corner of the triangle with the midpoint connecting the opposite side. The side bisectors, along with the vertical lines (line symmetries ), angle bisectors (angle symmetries) and the heights, belong to the classic transversals of triangular geometry .

properties

The bisector divides the triangular area into two triangles of the same height with respect to the common base and thus also the same area. By means of shear parallel to the bisector, the two partial triangles can be converted into an axially symmetrical shape while maintaining their area. This shear leaves the distribution of the surface elements within the partial triangles and thus the torque of the individual triangular surfaces related to the common base side unchanged. The three bisectors of a triangle are thus median lines and intersect at a point, the so-called center of gravity of the triangle. This divides each of the side bisectors in a ratio of 2: 1. The distance between the center of gravity and the corner is longer than the distance between the center of gravity and the center of the side.

The lengths of the side bisectors belonging to side a, b and c are calculated with:

${\ displaystyle s_ {a} = {\ frac {\ sqrt {2 (b ^ {2} + c ^ {2}) - a ^ {2}}} {2}}}$

${\ displaystyle s_ {b} = {\ frac {\ sqrt {2 (c ^ {2} + a ^ {2}) - b ^ {2}}} {2}}}$

${\ displaystyle s_ {c} = {\ frac {\ sqrt {2 (a ^ {2} + b ^ {2}) - c ^ {2}}} {2}}}$

Medians in tetrahedra

Medians of a tetrahedron with a focus on S.

{\ displaystyle {\ begin {aligned} & {\ frac {| AS |} {| SS_ {BCD} |}} = {\ frac {| BS |} {| SS_ {ACD} |}} = {\ frac { | CS |} {| SS_ {ABD} |}} \\ = & {\ frac {| DS |} {| SS_ {BBC} |}} = {\ frac {3} {1}} \ end {aligned} }}

In a tetrahedron , a line that connects a corner point with the center of gravity of the triangular surface opposite the corner point is called the median of the tetrahedron. The four medians of a tetrahedron intersect at one point, the center of gravity of the tetrahedron. This divides the medians in a ratio of 3: 1 ( theorem of Commandino ).

literature

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. 63
Harald Scheid , Wolfgang Schwarz: Elements of geometry . 5th edition. Springer, 2016, ISBN 978-3-662-50323-2 , p. 21
Rolf Baumann: More success in mathematics: 8th grade geometry . Mentor, 2008, ISBN 978-3-580-65629-4 , p. 29

Web links

Wiktionary: bisector - explanations of meanings, origins of words, synonyms, translations

Commons : bisector - collection of images, videos and audio files

Eric W. Weisstein : Triangle Median . In: MathWorld (English).
Derivation of formulas for the focus on the triangle

Individual evidence

^ ^A ^b Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical journeys of discovery . Springer, 2015, ISBN 978-3-662-45461-9 , p. 63
^ Claudi Alsina, Roger B. Nelsen: A Mathematical Space Odyssey: Solid Geometry in the 21st Century . The Mathematical Association of America, 2015, ISBN 978-0-88385-358-0 , pp. 97-98

[AN-1] A ^b Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical journeys of discovery . Springer, 2015, ISBN 978-3-662-45461-9 , p. 63

[2] Claudi Alsina, Roger B. Nelsen: A Mathematical Space Odyssey: Solid Geometry in the 21st Century . The Mathematical Association of America, 2015, ISBN 978-0-88385-358-0 , pp. 97-98