# When Four Cyclic Antipodal Pairs Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry

## Abstract

**:**

## 1. Introduction

- (1)
- (2)
- It is shown in [1] that if viewed in the relativistic model of hyperbolic geometry, the half-gyroangled gyrotrigonometric identity (1) gives rise to the novel hyperbolic Ptolemy’s Theorem in the relativistic model of hyperbolic geometry. Remarkably, the relativistic model of hyperbolic geometry, in turn, is identical to the well-known Klein model of hyperbolic geometry [3].
- (3)

- (1)
- within the framework of trigonometry, as a half-angled trigonometric identity in Euclidean geometry;
- (2)
- within the framework of gyrotrigonometry, as a half-gyroangled gyrotrigonometric identity in hyperbolic geometry.

## 2. Four Cyclic Antipodal Pairs

**Theorem 1.**

**(A Four Cyclic Antipodal Pairs Theorem).**Let $\Sigma (O,r)$ be a circle in the Euclidean plane ${\mathbb{R}}^{2}$ with radius r, centered at the origin $O\in {\mathbb{R}}^{2}$, with four cyclic antipodal pairs $(A,{A}^{\prime})$, $(B,{B}^{\prime})$, $(C,{C}^{\prime})$ and $(D,{D}^{\prime})$, such that the eight points $A,B,C,D,{A}^{\prime},{B}^{\prime},{C}^{\prime},{D}^{\prime}\in {\mathbb{R}}^{2}$ are arbitrarily ordered counterclockwise (or clockwise), as shown in Figure 1.

**Proof.**

## 3. Special Cases

**Corollary 1.**

**(A Three Cyclic Antipodal Pairs Theorem)**. Let $\Sigma (O,r)$ be a circle in the Euclidean plane ${\mathbb{R}}^{2}$ with radius r, centered at $O\in {\mathbb{R}}^{2}$, with three cyclic antipodal pairs $(A,{A}^{\prime})$, $(B,{B}^{\prime})$ and $(C,{C}^{\prime})$. The six points $A,B,C,{A}^{\prime},{B}^{\prime},{C}^{\prime}\in {\mathbb{R}}^{2}$ are arbitrarily ordered counterclockwise (or clockwise), as shown in Figure 2.

**Proof.**

## 4. Einstein Addition, Gyrogroups, Gyrovector Spaces and the Relativistic Model of Hyperbolic Geometry

## 5. The Law of Gyrocosines in the Relativistic Model of Hyperbolic Geometry

## 6. Four Gyrocyclic Antipodal Pairs

**Theorem 2.**

**(A Four Gyrocyclic Antipodal Pairs Theorem)**. Let $\Sigma (O,r)$ be a gyrocircle in the hyperbolic plane $({\mathbb{R}}_{s}^{2},\oplus ,\otimes )$ with gyroradius r, centered at $O\in {\mathbb{R}}_{s}^{2}$, with four gyrocyclic antipodal pairs $(A,{A}^{\prime})$, $(B,{B}^{\prime})$, $(C,{C}^{\prime})$ and $(D,{D}^{\prime})$, such that the eight points $A,B,C,D,{A}^{\prime},{B}^{\prime},{C}^{\prime},{D}^{\prime}\in {\mathbb{R}}_{s}^{2}$ are arbitrarily ordered counterclockwise (or clockwise), as shown in Figure 1 (viewed in the hyperbolic plane).

## 7. Three Gyrocyclic Antipodal Pairs

**Corollary 2.**

**(A Three Gyrocyclic Antipodal Pairs Theorem)**. Let $\Sigma (O,r)$ be a gyrocircle in the relativistic model of the hyperbolic plane ${\mathbb{R}}_{s}^{2}$ with gyroradius r, centered at $O\in {\mathbb{R}}_{s}^{2}$, with three gyrocyclic antipodal pairs $(A,{A}^{\prime})$, $(B,{B}^{\prime})$ and $(C,{C}^{\prime})$. The six points $A,B,C,{A}^{\prime},{B}^{\prime},{C}^{\prime}\in {\mathbb{R}}_{s}^{2}$ are arbitrarily ordered counterclockwise (or clockwise), as shown in Figure 2 (viewed in the hyperbolic plane).

## 8. The Pythagorean Theorem Is Recovered

## 9. A Hyperbolic Pythagorean Theorem

**Theorem 3.**

**(A Hyperbolic Pythagorean Theorem)**. Let $AB{B}^{\prime}$ be a gyrotriangle of which the side $B{B}^{\prime}$ coincides with a gyrodiameter of its circumgyrocircle in the relativistic model of the hyperbolic plane (Figure 1). Then,

## 10. Conclusions

- (1)
- we realize the half-angled trigonometric identity (1) within the framework of Euclidean geometry, obtaining the famous Ptolemy’s Theorem in Euclidean geometry;
- (2)
- we realize the same identity (1) as a gyrotrigonometric identity within the framework of hyperbolic geometry, obtaining the novel hyperbolic Ptolemy’s Theorem.

- (1)
- (2)
- In Section 6 and Section 7, we explore the realization of the same half-angled trigonometric identity (4) in hyperbolic geometry, obtaining Theorem 2; Section 4 and Section 5 present the preliminaries necessary for the transition from the Euclidean geometry in Section 2 to the hyperbolic geometry in Section 6;
- (3)

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Four cyclic antipodal pairs, $(A,{A}^{\prime})$, $(B,{B}^{\prime})$, $(C,{C}^{\prime})$ and $(D,{D}^{\prime})$, on a circle $\Sigma (O,r)$ centered at the origin $O=(0,0)\in {\mathbb{R}}^{2}$ of the Euclidean plane, with radius r, and their corresponding O-vertex angles $\alpha ,\phantom{\rule{3.33333pt}{0ex}}\beta ,\phantom{\rule{3.33333pt}{0ex}}\gamma ,\phantom{\rule{3.33333pt}{0ex}}\delta $. The points $A,B,C,D,{A}^{\prime},{B}^{\prime},{C}^{\prime},{D}^{\prime}$ are arbitrarily ordered counterclockwise, implying $\alpha +\beta +\gamma +\delta =\pi $. The identity of Theorem 1 is shown, where $\left|AB\right|=\parallel -A+B\parallel $, etc.

**Figure 2.**Three cyclic antipodal pairs, $(A,{A}^{\prime})$, $(B,{B}^{\prime})$ and $(C,{C}^{\prime})$, on a circle $\Sigma (O,r)$ centered at the origin $O=(0,0)\in {\mathbb{R}}^{2}$ of the Euclidean plane, with radius r. The points $A,B,C,{A}^{\prime},{B}^{\prime},{C}^{\prime}$ are arbitrarily ordered counterclockwise. The identities of Corollary 1 are shown, where $\left|AB\right|=\parallel -A+B\parallel $, etc.

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**MDPI and ACS Style**

Ungar, A.A.
When Four Cyclic Antipodal Pairs Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry. *Symmetry* **2024**, *16*, 729.
https://doi.org/10.3390/sym16060729

**AMA Style**

Ungar AA.
When Four Cyclic Antipodal Pairs Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry. *Symmetry*. 2024; 16(6):729.
https://doi.org/10.3390/sym16060729

**Chicago/Turabian Style**

Ungar, Abraham A.
2024. "When Four Cyclic Antipodal Pairs Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry" *Symmetry* 16, no. 6: 729.
https://doi.org/10.3390/sym16060729