Theorem of the central parallels in a triangle

The set of the agents parallels the triangle is a theorem from the mathematical sub-area of the triangle geometry . The theorem deals with an elementary property of the triangles of the Euclidean plane .

Central parallel in the triangle

Formulation of the sentence

The sentence says the following:

In a triangle of the Euclidean plane, the line connecting the centers of two sides is always parallel to the third side of the triangle and always half as long as it.

proof

The theorem is based on elementary geometry , with the assertion of parallelism following from the inversion of the first theorem of rays , while the statement about the length ratio is then obtained with the second theorem of rays.

Another proof using vector calculus goes as follows:

Based on the definition (see picture) that the triangle has the corner points and that the center is the side and the center of the side , you set ${\displaystyle A,B,C}$ ${\displaystyle D}$ ${\displaystyle [AB]}$ ${\displaystyle E}$ ${\displaystyle [AC]}$

${\displaystyle {\vec {a}}={\overrightarrow {BC}}\;,\;{\vec {b}}={\overrightarrow {CA}}\;,\;{\vec {c}}={\overrightarrow {AB}}\;,\;{\vec {m}}_{a}={\overrightarrow {DE}}}$   .

The equations are obtained from this

${\displaystyle {\frac {1}{2}}{\vec {b}}={\overrightarrow {CE}}={\overrightarrow {EA}}\;,\;{\frac {1}{2}}{\vec {c}}={\overrightarrow {AD}}={\overrightarrow {DB}}}$   .

So follows

${\displaystyle {\vec {m}}_{a}+{\frac {1}{2}}{\vec {b}}+{\frac {1}{2}}{\vec {c}}={\vec {0}}}$

such as

${\displaystyle {\vec {m}}_{a}-{\frac {1}{2}}{\vec {b}}-{\vec {a}}-{\frac {1}{2}}{\vec {c}}={\vec {0}}}$   .

Then adding the left and right sides of the last two equations gives

${\displaystyle 2{\vec {m}}_{a}-{\vec {a}}={\vec {0}}}$

and thus

${\displaystyle {\vec {m}}_{a}={\frac {1}{2}}{\vec {a}}}$   .

This means, on the one hand, that the two straight lines on which the lines or lie, run in the same direction and are therefore parallel, and, on the other hand, that the lengths of the two lines and the alleged relationship, namely ${\displaystyle [BC]}$ ${\displaystyle [DE]}$ ${\displaystyle [BC]}$ ${\displaystyle [DE]}$

${\displaystyle {\overline {DE}}={\frac {1}{2}}{\overline {BC}}}$

fulfill.

For the other two central parallels the proof goes accordingly.

Sources and literature

• IN Bronstein , KA Semendjajev , G. Musiol , H. Mühlig (Hrsg.): Taschenbuch der Mathematik . 7th, completely revised and expanded edition. Verlag Harri Deutsch , Frankfurt am Main 2008, ISBN 978-3-8171-2007-9 .
• Siegfried Krauter : Experience elementary geometry . A workbook for independent and active discovery. Spektrum Akademischer Verlag , Munich 2005, ISBN 3-8274-1644-2 .
• Wilhelm Kuypers , Josef Lauter (Ed.): Mathematics Secondary Level II . Analytical Geometry and Linear Algebra. Cornelsen Verlag , 1992.

References and footnotes

1. ^ Siegfried Krauter: Experience elementary geometry. 2005, p. 62
2. ↑ ^{a b} Wilhelm Kuypers, Josef Lauter (Hrsg.): Mathematics secondary level II. Analytical geometry and linear algebra. 1992, p. 40
3. ^ IN Bronstein, KA Semendjajev et al .: Taschenbuch der Mathematik. 2008, p. 136
4. ↑ Compare the triangle with the corner points in the picture ! ${\displaystyle D,E,A}$
5. ↑ Compare the square with the corner points in the picture ! ${\displaystyle D,E,C,B}$