Triangle inscribed in another triangle

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Triangle inscribed in another triangle

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Source: Cono Sur 2010 #5

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#1 h Nov 17, 2015, 2:37 AM • 2 Y

Y by Adventure10, Mango247

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 $ABC$ .

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#2 h Nov 25, 2015, 3:18 AM • 3 Y

Y by NiltonCesar, Adventure10, Mango247

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 $r_C \equiv XY$ $\Longrightarrow$ $\triangle (r_A,r_B,r_C) \equiv \triangle XYZ.$

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