Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

## Angle and Triangle in Euclidian Topological Space

Akihiro Kubo
Shinshu University, Nagano
Yatsuka Nakamura
Shinshu University, Nagano

### Summary.

Two transformations between the complex space and 2-dimensional Euclidian topological space are defined. By them, the concept of argument is induced to 2-dimensional vectors using argument of complex number. Similarly, the concept of an angle is introduced using the angle of two complex numbers. The concept of a triangle and related concepts are also defined in \$n\$-dimensional Euclidian topological spaces.

#### MML Identifier: EUCLID_3

The terminology and notation used in this paper have been introduced in the following articles [17] [20] [19] [21] [3] [13] [22] [4] [8] [18] [12] [5] [14] [16] [9] [2] [6] [7] [1] [11] [10] [15]

Contents (PDF format)

#### Bibliography

[1] Kanchun   and Yatsuka Nakamura. The inner product of finite sequences and of points of \$n\$-dimensional topological space. Journal of Formalized Mathematics, 15, 2003.
[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. The complex numbers. Journal of Formalized Mathematics, 2, 1990.
[6] Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.
[7] Wenpai Chang and Yatsuka Nakamura. Inner products and angles of complex numbers. Journal of Formalized Mathematics, 15, 2003.
[8] Library Committee. Introduction to arithmetic. Journal of Formalized Mathematics, Addenda, 2003.
[9] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[10] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[11] Agata Darmochwal and Yatsuka Nakamura. The topological space \$\calE^2_\rmT\$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[12] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[13] Jaroslaw Kotowicz. Real sequences and basic operations on them. Journal of Formalized Mathematics, 1, 1989.
[14] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[15] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[16] Agnieszka Sakowicz, Jaroslaw Gryko, and Adam Grabowski. Sequences in \$\calE^N_\rmT\$. Journal of Formalized Mathematics, 6, 1994.
[17] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[18] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[19] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[20] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[22] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Journal of Formalized Mathematics, 10, 1998.

Received May 29, 2003

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