
06340552
j
2015a.00609
Garc\'{\i}a, Emmanuel Antonio Jos\'e
A note on reflections.
Forum Geom. 14, 155161 (2014).
2014
Florida Atlantic University, Department of Mathematical Sciences, Boca Raton, FL
EN
G45
reflections on triangles
elementary geometry
Zbl 0294.50010
http://forumgeom.fau.edu/FG2014volume14/FG201414index.html
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 $P$.
Wolfgang Globke (Adelaide)