Background and aim: Monitoring bioelectric phase angle (PhA) provides important information on the health and the condition of the athlete. Together with the vector length, PhA constitutes the bioimpedance vector analysis (BIVA) patterns, and their joint interpretation exceeds the limits of the evaluation of the PhA alone. The present investigation aimed to monitor changes in the BIVA patterns during a training macrocycle in swimmers, trying to ascertain if these parameters are sensitive to training load changes across a 13-week training period. Methods: Twelve national and international level swimmers (four females; eight males; 20.9 ± 1.9 years; with a competitive swimming background of 11.3 ± 1.8 years; undertaking 16–20 h of pool training and 4–5 h of dry-land training per week and 822.0 ± 59.0 International Swimming Federation (FINA) points) were evaluated for resistance (R) and reactance (Xc) using a single frequency phase sensitive bioimpedance device at the beginning of the macrocycle (M1), just before the beginning of the taper period (M2), and just before the main competition of the macrocycle (M3). At the three-time assessment points, swimmers also performed a 50 m all-out first stroke sprint with track start (T50 m) while time was recorded. Results: The results of the Hotelling T2 test showed a significant vector displacement due to simultaneous R and Xc changes (p < 0.001), where shifting from top to bottom along the major axis of the R-Xc graph from M1 to M2 was observed. From M2 to M3, a vector displacement up and left along the minor axis of the tolerance ellipses resulted in an increase in PhA (p < 0.01). The results suggest a gain in fluid with a decrease in cellular density from M1 to M2 due to decrements in R and Xc. Nevertheless, the reduced training load characterizing taper seemed to allow for an increase in PhA and, most importantly, an increase of Xc, thus demonstrating improved cellular health and physical condition, which was concomitant with a significant increase in the T50 m performance (p < 0.01). Conclusions: PhA, obtained by bioelectrical R and Xc, can be useful in monitoring the condition of swimmers preparing for competition. Monitoring BIVA patterns allows for an ecological approach to the swimmers’ health and condition assessment without resorting to equations to predict the related body composition variables.

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The area method for Euclidean constructive geometry was proposed by Chou, Gao and Zhang in the early 1990's. The method can efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated theorem proving in geometry. The method produces proofs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted in the original presentations. We also give a variant of Chou, Gao and Zhang's axiom system. Based on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more. Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of details that has to be dealt with. With the description of the method given in this paper, implementing the method should be still complex, but a straightforward task. In the paper we describe all these implementations and also some of their applications. 490 P. Janičić et al.

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