In this paper we study the generalization of the Brocard congurations. The denition of q-period broken line is given and then the new necessary and sucient condition about the existence of two conjugate Brocard points of the q􀀀period loop broken line is investigated. Based on the past research, a geometric method for the construction of all possible loop broken lines with conjugate Brocard points is developed.

Forums have been one of the most important internet services since the 21st century. However, Forum users have to receive information in a passive way currently. The topics in forums are ordered by last update time (last reply time), thus the information that a user is interested in may be overwhelmed by a large number of other information. Users always have to scan many pages to find a minority of information they need. In this paper, based on the analysis of users' needs, I have designed a personalized forum topic ranking system. This personalized ranking system first calculates all the factors that will influence the user’s decision-making as to whether or not to view a topic by using his/her browsing history. Then the system predicts the click probability for each topic according to all the influence factors using a learned maximum entropy model. Finally, forum topics can be ranked by the predicted click probability, so as to make users find their favorite information easier. As shown by the experiments, the precision of the personalized ranking system is about 60% to 75%, which improves the traditional method (ordered by last update time) by 50% to 85%. In addition, with the normalization of the indicator functions of the maximum entropy model and the selection of the iterative endpoint, the training time can be lowered to an average of 0.01 seconds for each user. It indicates that such a model is able to meet the requirements of practical applications.

In the early 1980s of twenty century, Professor Zhang Zhongfu, an expert in graph theory, raised a question in his research: how many points of intersection of diagonal lines are there inside a regular N polygon, hoping to find out a formula of count. It has been more than 20 years since the question was raised, which has aroused the interest of quite a lot of experts, scholars and those who love mathematics, but still remains unsolved. When N is an odd number, proposition 1 can be derived from formula of counting based on proposition 2, which can also be verified by the programme the author has made. Proposition 1: when N is an odd number, there is no concurrence of 3 or more than 3 diagonal lines of regular N polygon. Proposition 2: when N is an odd number, the number of intersection points of inner diagonal lines of regular N polygon is: &a_n=c^{4}_{n}=frac{1}{24}(n-1)(n-2)(n-3)$ But when N is an even number, it becomes quite complicated. When N are some special even numbers referred to by Chang Jianming, there are some laws: the research is made from the perspective of prime factorization of figures. The writer primarily studies the issue by making use of geometrical drawing and imitation by computer programme, so that the programme is improved and the imitating solutions of the numbers of intersection points of inner diagonal lines of regular N polygon with specific programmes can be solved. The solution is popularizable. The writer has worked out the numbers of intersection points of inner diagonal lines of regular 2N polygon when N 50, if conditions of computers in schools allow. By comparing the counting between drawing and computer imitation, the writer has also found out some interesting laws about the numbers of intersection points of inner diagonal lines of regular N polygon, and has put forward some hypotheses.

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