# On Some Properties of the First Brocard Triangle in the Isotropic Plane

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## Abstract

**:**

## 1. Introduction and Motivation

## 2. The First Brocard Triangle of a Triangle in the Isotropic Plane

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Proof.**

## 3. The First Brocard Triangle and Some Other Significant Elements

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Theorem**

**9.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Theorem**

**13.**

**Proof.**

**Theorem**

**14.**

**Proof.**

**Theorem**

**15.**

**Proof.**

**Theorem**

**16.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Brocard circle ${\mathcal{K}}_{b}$, the Steiner axis $\mathcal{S}$, and the first Brocard triangle ${A}_{1}{B}_{1}{C}_{1}$ of the triangle $ABC$. Visualization of statements of Theorems 1, 3 and 4.

**Figure 2.**The axis of homology $\mathcal{T}$ of the triangle $ABC$ and its first Brocard triangle ${A}_{1}{B}_{1}{C}_{1}$, and the Kiepert parabola $\mathcal{P}$ of the triangle $ABC$. Visualization of statements of Theorems 5 and 7.

**Figure 3.**The circumscribed Steiner ellipse ${\mathcal{S}}_{e}$ of the triangle $ABC$. Visualization of statements of Theorems 8–11.

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**MDPI and ACS Style**

Volenec, V.; Kolar-Begović, Z.; Kolar-Šuper, R. On Some Properties of the First Brocard Triangle in the Isotropic Plane. *Mathematics* **2022**, *10*, 1381.
https://doi.org/10.3390/math10091381

**AMA Style**

Volenec V, Kolar-Begović Z, Kolar-Šuper R. On Some Properties of the First Brocard Triangle in the Isotropic Plane. *Mathematics*. 2022; 10(9):1381.
https://doi.org/10.3390/math10091381

**Chicago/Turabian Style**

Volenec, Vladimir, Zdenka Kolar-Begović, and Ružica Kolar-Šuper. 2022. "On Some Properties of the First Brocard Triangle in the Isotropic Plane" *Mathematics* 10, no. 9: 1381.
https://doi.org/10.3390/math10091381