Monge point

The Monge point is an object of spatial geometry . It is named after the French mathematician Gaspard Monge , who was the first to describe this excellent point of the general tetrahedron and to characterize it by Monge's theorem presented below .

Sentence and definition

Given a tetrahedron with edges . For each edge, let the respective center point and the opposite edge be. There is exactly one plane through each such that and are exactly perpendicular to one another. ${\displaystyle {\mathcal {T}}\subset \mathbb {R} ^{3}}$ ${\displaystyle a,\,b,\,c,\,d,\,e,\,f}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle x}$ ${\displaystyle M(x)}$ ${\displaystyle x^{'}}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle M(x)}$ ${\displaystyle {\mathcal {E}}(x)\subset \mathbb {R} ^{3}}$ ${\displaystyle {\mathcal {E}}(x)}$ ${\displaystyle x^{'}}$

The following applies:

The average consists of exactly one point . ${\displaystyle \textstyle \bigcap _{x=a,b,c,d,e,f}{{\mathcal {E}}(x)}}$ ${\displaystyle M({\mathcal {T}})\in {\mathcal {T}}}$

This clearly defined point is the Monge point of . ${\displaystyle M({\mathcal {T}})\in {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$

The levels described above are also called Monge-levels (Engl. Monge plan called). With these, Monge's theorem can be reproduced in a nutshell as follows: ${\displaystyle {\mathcal {E}}(x)\subset \mathbb {R} ^{3}}$   ${\displaystyle {(x=a,b,c,d,e,f)}}$

In a tetrahedron , the Monge planes intersect at one point, namely at the Monge point . ${\displaystyle {\mathcal {T}}\subset \mathbb {R} ^{3}}$ ${\displaystyle M({\mathcal {T}})\in {\mathcal {T}}}$

The Mannheim theorem

The following theorem, which goes back to the French mathematician Amédée Mannheim , can also be used to characterize the Monge point :

If one places the (clearly defined!) Plane in the tetrahedron through each of its four heights as well as the height intersection of the perpendicular side triangle associated with the respective height, then the four levels given in this way have the Monge point as the intersection . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle M({\mathcal {T}})}$

Position on Euler's straight line

In general tetrahedron is Euler line (engl. Euler line ) that is precisely what the focus of and the center of the circumscribed sphere of connects. The Monge point turns out to be that particular point of the general tetrahedron , which lies on the straight line as a mirror image of the point . In other words: The Monge point is generally a tetrahedron on the straight line beyond such that the midpoint of the segment is. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle e({\mathcal {T}})\subset \mathbb {R} ^{3}}$ ${\displaystyle S({\mathcal {T}})}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle Z({\mathcal {T}})}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle M({\mathcal {T}})}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle S({\mathcal {T}})}$ ${\displaystyle Z({\mathcal {T}})}$ ${\displaystyle e({\mathcal {T}})}$ ${\displaystyle M({\mathcal {T}})}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle e({\mathcal {T}})}$ ${\displaystyle S({\mathcal {T}})}$ ${\displaystyle S({\mathcal {T}})}$ ${\displaystyle {\overline {{M({\mathcal {T}})}{Z({\mathcal {T}})}}}}$

literature

items

• HF Thompson : A Geometrical Proof of a Theorem connected with the Tetrahedron . In: Proceedings of the Edinburgh Mathematical Society. (Series I) . tape 27 , 1908, pp. 51-53 .

Monographs

• Nathan Altshiller-Court: Modern Pure Solid Geometry . 2nd Edition. Chelsea Publishing Company, Bronx, NY 1964, OCLC 1597161 .
• Howard Eves: An Introduction to the History of Mathematics . 5th edition. Saunders College Publishing, Philadelphia [et al. a.] 1983, ISBN 0-03-062064-3 .
• Heinrich Schröter: Theory of the surfaces of the second order and the space curves of the third order as products of projective structures . Teubner, Leipzig 1880.

Individual evidence

1. ^ Nathan Altshiller-Court: Modern Pure Solid Geometry. 1964, pp. 76, 340
2. ^ ^{A b} Heinrich Schröter: Theory of the surfaces of the second order and the space curves of the third order as products of projective structures. 1880, p. 209
3. ↑ HF Thompson: A Geometrical Proof of a Theorem connected with the Tetrahedron. 1908, p. 51
4. ^ ^{A b} Nathan Altshiller-Court: Modern Pure Solid Geometry. 1964, p. 77
5. ^ Nathan Altshiller-Court: Modern Pure Solid Geometry. 1964, pp. 78-79
6. ^ Howard Eves: An Introduction to the History of Mathematics. 1983, p. 340