# Brocardove točke i Broacardov kut

Klisura, Tajana (2016) Brocardove točke i Broacardov kut. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

Brocard point is one of the many special points of a triangle. It is defined as a point $\Omega$ inside a triangle ABC such that line segments $\bar{PA}, \bar{PB}$ and $\bar{PC}$ form congruent angles with sides $\bar{AB}, \bar{BC}$ and $\bar{CA}$, respectively. Corresponding congruent angles are called Brocard angle. By reversing the order of vertices we obtain second Brocard point. It is possible to generalize the concept of the Brocard point to n-polygons. As opposed to the case of a triangle, where a unique Brocard point always exists, for n-polygons where $n \geq 4$a point and an angle with corresponding properties may exist, but not necessarily. In the first chapter we explain the basic, well-known results on the Brocard point and Brocard angle of a triangle: proofs of existence by construction, the upper bound for Brocard angle and the formula $ctg \omega = ctg \alpha + ctg \beta + ctg \gamma$; where $\alpha, \beta, \gamma$ are angles of a triangle. In the second chapter we present some results about the Brocard point and the Brocard angle of a general polygon. The existence of the Brocard point is investigated using a sequence of the polygons obtained by the so called Brocard transformation, that converges to a nonempty set, which is either a one point set or a line segment. A Criterion for the existence of the Brocard point is proven, expressed in terms of similarity between the given polygon and any of its Brocard transforms. In the case of existence of the Brocard point the stability property is also shown, meaning that all the Brocard transforms have the same Brocard point. One of the main results proven for $n$-polygons which have the Brocard point is the estimate of the Brocard angle $\omega$, showing that $\omega \leq \frac{\pi}{2} - \frac{\pi}{n}$ where equality holds only for a regular polygon. It was recently noticed that this estimate easily follows from a much more general result by Dmitriev and Dynkin (1945.). This chapter ends with example of irregular polygons for which the Brocard points exist.