1. In a triangle $ABC$, $O$ is the circumcenter and $I$ is the incenter. $X$ is the reflection of $I$ to $O$. $A_1$ is foot of the perpendicular from $X$ to $BC$. $B_1$ and $C_1$ are defined similarly. prove that $AA_1$,$BB_1$ and $CC_1$ are concurrent.(12 points)
Problem
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Tags: geometry, circumcircle, geometric transformation, reflection, geometry proposed
mahanmath
07.08.2010 14:22
Hint : $1)$ Use Ceva Theorem $2)$ $ BA_1 = p-c$
vladimir92
07.08.2010 15:07
goodar2006 wrote: 1. In a triangle $ABC$, $O$ is the circumcenter and $I$ is the incenter. $X$ is the reflection of $I$ to $O$. $A_1$ is foot of the perpendicular from $X$ to $BC$. $B_1$ and $C_1$ are defined similarly. prove that $AA_1$,$BB_1$ and $CC_1$ are concurrent.(12 points) thank's for that easy problem and nice problem! I think I follow the same as mahanmath.
Project $O$ and $I$ on $BC$ at $O_A$ and $I_A$, and let incircle touche $AC$ and $AB$ at $I_B$ and $I_C$, then $O_A$ is midpoint of $BC$ and $A_1I_A$ it follow that $BI_A=CA_1$, then $AA_1$,$BB_1$ and $CC_1$ are concurrent if $AI_A$ ,$BI_B$ and $CI_C$ are concurrent, which is true since those last concuer at Gregonne point.
jayme
07.08.2010 15:14
Dear Mathlinkers, X is the Bevan's point of ABC. For more explications see http://perso.orange.fr/jl.ayme , vol. 3 Cinq théorèmes de von Nagel, p. 22. Sincerely Jean-Louis