Olym 3G. 닮음과 Mix Circles
- 실전문제
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Let $ABC$ be a triangle and $\Omega$ its circumcircle. Let the internal angle bisectors of $\angle BAC, \angle ABC, \angle BCA$ intersect $BC,CA,AB$ on $D,E,F$, respectively. The perpedincular line to $EF$ through $D$ intersects $EF$ on $X$ and $AD$ intersects $EF$ on $Z$. The circle internally tangent to $\Omega$ and tangent to $AB,AC$ touches $\Omega$ on $Y$. Prove that $(XYZ)$ is tangent to $\Omega$.
Olympic Revenge
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Let $\triangle ABC$ be an acute triangle, let $\Gamma$ be its circumcircle and $O$ its circumcenter. Let $M,E,F$ be the midpoints of sides $BC,AC,AB$. Let $\omega$ be the circle through $E,F$ and tangent to $\Gamma$ at $T\neq A$. Let $AT\cap \omega=R$.
Prove $ROMT$ is cyclic.
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