Synthetic foundations of cevian geometry, III: The generalized orthocenter

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE, QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map $\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.Comment: 34 pages, 7 figure

Synthetic foundations of cevian geometry, I: Fixed points of affine maps in triangle geometry

We give synthetic proofs of many new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangle $DEF$ of a point $P$ with respect to a given triangle $ABC$, as well as the cevian triangle of the isotomic conjugate $P'$ of $P$ with respect to $ABC$. We prove a formula for the cyclocevian map in terms of the isotomic and isogonal maps using an entirely synthetic argument, and show that the complement $Q$ of the isotomic conjugate $P'$ has many interesting properties. If $T_P$ is the affine map taking $ABC$ to $DEF$, we show synthetically that $Q$ is the unique ordinary fixed point of $T_P$ when $P$ is any point not lying on the sides of triangle $ABC$, its anti-complementary triangle, or the Steiner circumellipse of $ABC$. We also show that $T_P(Q')=P$ if $Q'$ is the complement of $P$, and that the affine map $T_P T_{P'}$ is either a homothety or a translation which always has the $P$-ceva conjugate of $Q$ as a fixed point. Finally, we show that $P$ lies on the Steiner circumellipse if and only if $T_PT_{P'}=K^{-1}$, where $K$ is the complement map for $ABC$. This paper forms the foundation for several more papers to follow, in which the conic on the 5 points $A,B,C,P,Q$ is studied and its center is characterized as a fixed point of the map $\lambda=T_{P'} T_P^{-1}$.Comment: 25 pages, 6 figure

Vertex Positions of the Generalized Orthocenter and a Related Elliptic Curve

We study triangles $ABC$ and points $P$ for which the generalized orthocenter $H$ corresponding to $P$ coincides with a vertex $A,B$, or $C$. The set of all such points $P$ is a union of three ellipses minus $6$ points. In addition, if $T_P$ is the affine map taking $ABC$ to the cevian triangle $DEF$ of $P$ with respect to $ABC$, $P'$ is the isotomic conjugate of $P$, and $T_{P'}$ is the affine map taking $ABC$ to the cevian triangle of $P'$, then we study the locus of points $P$ for which the map $\textsf{M}_P=T_p \circ K^{-1} \circ T_{P'}$ is a translation. Here, $K$ is the complement map for $ABC$, and $\textsf{M}_P$ is an affine map taking the circumconic of $ABC$ for $P$ to the inconic of $ABC$ for $P$. The locus in question turns out to be an elliptic curve minus $6$ points, which can be synthetically constructed using the geometry of the triangle

Wise Therapists Say: A Qualitative Research Study

The purpose of this study was to investigate the perspectives of experienced music therapists regarding their career longevity in the field. This study further aimed to determine if themes relating to career longevity in music therapy would emerge when gathering perspectives of experienced music therapists. Modified grounded theory was used to examine semi-structured interviews with four music therapists with 20 or more years of experience. The participants consisted of four music therapists drawn from members of the Canadian Association of Music Therapists. The results of the study were divided into four categories: The surveyed music therapists’ educational background and motivations to become music therapists; music therapist participant’s reported sources of struggle throughout their music therapy careers; music therapist’s perceived important factors contributing to their own career longevity; and music therapist participants’ advice to emerging music therapists, with descriptive statements describing career longevity among music therapists as the final product. Results of the study suggest that self-care, adaptability, advocacy, and commitment to music therapy as a unique mode of intervention were considered significant to career longevity in music therapy practice. Interpretations as well as potential implications for practice, education and research are discussed

Values of Youth as a Resource for Developing the Company’s Internal Corporate Ideology: Globalization and Patriotism

Corporate culture as it exists today is a cast of the Western European matrix, in which the ideology of the state is replaced by the values of the corporation. But modern realities form a new demand of society, moral principles and traditions develop modern legal relations.The purpose of the article (in the form of confirmation or refutation of a scientific hypothesis) is to analyze the value system of youth, legal realities and ideological features of the formation of the Russian state around the factors of Russian identity: family, society, country, state, person, to describe the possibility of their inclusion in the corporate culture (employer) as the basic principles of corporate governance.The group interview method and the content analysis method were chosen as the research method. The method of group interview allows each of the participants of the discussion to speak freely and the participants of the debate to form a unified representation of the group on the problems of corporate culture, which are based on the factors of Russian identity: family, person, society, country, state. The content analysis method allows, based on recorded discussions, to identify the key values of young people that can be included in the corporate culture of Russian companies

Вирусно-бактериальная инфекция и коронарный атеросклероз

Currently diseases of cardiovascular system rank the leading place in the structure of mortality of the population in industrialized countries. The basis of this nosology is atherosclerotic vascular lesions. Active search for the etiological factors, initiating the development of atherosclerosis, leads to new hypotheses. However, today any of declared is not aprioristic. Chronic inflammation supported by persistency of Chlamidia pneumoniae, Mycoplasma pneumoniae, Cytomegalovirus, Herpes simplex types 1 and 2 promotes activation of the endothelium, leading to its metabolic parameter changes that influence the lipid metabolism, and thus participates in the development of atherosclerotic lesions.В настоящее время ведущее место в структуре летальности населения промышленно развитых стран занимают заболевания сердечно-сосудистой системы [1]. В основе этой нозологии лежит атеросклеротическое поражение сосудов [2]. Активный поиск этиологического фактора, инициирующего развитие атеросклероза, приводит к появлению новых гипотез. Однако на сегодняшний день ни одна из заявленных не является априорной. Хронический воспалительный процесс, поддерживаемый персистирующей инфекцией Chlamidia pneumoniae, Mycoplasma pneumoniae, Cytomegalovirus, Herpes simplex 1-го и 2-го типов способствует активации эндотелия, приводя к изменению его метаболических параметров, оказывает влияние на липидный обмен и таким образом участвует в развитии атеросклеротического поражения [3]