@article {MATHEDUC.06340552,
author = {Garc\'{\i}a, Emmanuel Antonio Jos\'e},
title = {A note on reflections.},
year = {2014},
journal = {Forum Geometricorum},
volume = {14},
issn = {1534-1178},
pages = {155-161},
publisher = {Florida Atlantic University, Department of Mathematical Sciences, Boca Raton, FL},
abstract = {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. Recall 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 $2\overline{SG}$. The 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 $S$. The 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 $P$. Now 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 $Q/I$. Finally, Garc\'ia reproves a result due to {\it 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 $P$.},
reviewer = {Wolfgang Globke (Adelaide)},
msc2010 = {G45xx},
identifier = {2015a.00609},
}