AoPS - Problem

Source: Cono Sur 2010 #5

Tags: cono sur, geometry proposed, geometry

fprosk

17.11.2015 04:37

The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D, E$, and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$, and $DCE$, respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$, respectively. Let $r_A$ be the line $E_{a}F_a$. Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$, $r_B$, and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$.

Let $X,Y,Z$ be the midpoints of $BC,CA,AB,$ respectively. If $BI$ cuts $DE$ at $U,$ we have $\angle BUD=\angle EDC-\angle IBC=90^{\circ}-\tfrac{1}{2}\angle ACB-\tfrac{1}{2}\angle ABC=\tfrac{1}{2}\angle BAC=\angle IAE$ $\Longrightarrow$ $U \in \omega_a$ $\Longrightarrow$ $U \equiv E_a$ $\Longrightarrow$ $\angle AE_aB \equiv \angle AE_aI=90^{\circ}$ $\Longrightarrow$ $\angle AZE_a=2\angle ABE_a=\angle ABC$ $\Longrightarrow$ $ZE_a \parallel BC$ $\Longrightarrow$ $E_a \in YZ$ and likewise $F_a \in YZ$ $\Longrightarrow$ $r_A \equiv YZ$ and similarly $r_B \equiv ZX$ and $r_C \equiv XY$ $\Longrightarrow$ $\triangle (r_A,r_B,r_C) \equiv \triangle XYZ.$