Olympiad problems

2020 Israel Olympic Revenge, P4

Tags: feuerbach , geometry

Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo: Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let $F_A$ be the (associated with $\Omega$) Feuerbach point of the triangle formed by the tangents to $\Omega$ at $B,C,D$, that is, the point of tangency of $\Omega$ and the nine-point circle of that triangle. Define $F_B, F_C, F_D$ similarly. Let $A'$ be the intersection of the tangents to $\Omega$ at $A$ and $F_A$. Define $B', C', D'$ similarly. Prove that quadrilaterals $ABCD$ and $A'B'C'D'$ are similar

KoMaL A Problems 2019/2020, A. 758

Tags: geometry , feuerbach

In a quadrilateral $ABCD,$ $AB=BC=DA/\sqrt{2},$ and $\angle ABC$ is a right angle. The midpoint of $BC$ is $E,$ the orthogonal projection of $C$ on $AD$ is $F,$ and the orthogonal projection of $B$ on $CD$ is $G.$ The second intersection point of circle $(BCF)$ (with center $H$) and line $BG$ is $K,$ and the second intersection point of circle $(BCF)$ and line $HK$ is $L.$ The intersection of lines $BL$ and $CF$ is $M.$ The center of the Feurbach circle of triangle $BFM$ is $N.$ Prove that $\angle BNE$ is a right angle. [i]Proposed by Zsombor Fehér, Cambridge[/i]

2019 IFYM, Sozopol, 8

Tags: inversion , feuerbach , geometry

We are given a $\Delta ABC$. Point $D$ on the circumscribed circle k is such that $CD$ is a symmedian in $\Delta ABC$. Let $X$ and $Y$ be on the rays $\overrightarrow{CB}$ and $\overrightarrow{CA}$, so that $CX=2CA$ and $CY=2CB$. Prove that the circle, tangent externally to $k$ and to the lines $CA$ and $CB$, is tangent to the circumscribed circle of $\Delta XDY$.