A note on reflections (Q2878866)

Given a triangle with vertices \(A,B,C\) and a point \(P\), let \(X,Y,Z\) be the reflections of \(P\) on the midpoints \(M_a, M_b,M_c\) of the sides \(BC,CA,AB\), respectively.NEWLINENEWLINERecall that the Euler line of a triangle is the line passing through it centroid and its orthocenter. Let \(G\) be the centroid of the triangle \(ABC\). The anticomplement of a point \(S\) (with respect to \(ABC\)) is the point \(S'\) which lies on the line \(SG\) on the opposite side of \(G\) with distance \(2\overline{SG}\).NEWLINENEWLINEThe first main theorem in this paper is that if the Euler lines of \(PBC\), \(PCA\) and \(PAB\) intersect in a point \(S\), then the Euler lines of \(AZY\), \(BXZ\) and \(CYX\) intersect in the anticomplement of \(S\).NEWLINENEWLINEThe cevian quotient \(Q/P\) of two points \(P,Q\) with respect to \(ABC\) is the perspective center mapping the cevian triangle of \(Q\) to the anticevian triangle of \(P\).NEWLINENEWLINENow let \(Q\) be the center of the incircle of the triangle \(M_a M_b M_c\). The second main theorem states that if \(P\) is the center of the incircle of \(ABC\), then the Euler lines of \(XBC\), \(YCA\) and \(ZAB\) intersect at the cevian quotient \(Q/I\).NEWLINENEWLINEFinally, García reproves a result due to \textit{S. N. Collings} [Math. Gaz. 58, 264 (1974; Zbl 0294.50010)], which states that the circles \((AYZ)\), \((BZX)\) and \((CXY)\) intersect in a point on the circumcircle of \(ABC\), which is the anticomplement of the center of the rectangular hyperbola through \(A,B,C\) and \(P\).