AoPS - Problem

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Problem

Source: V.A. Yasinsky Geometry Olympiad 2023 VIII p1 , Ukraine

Tags: geometry

parmenides51

12.12.2023 14:33

Let $O$ be the circumcenter of triangle $ABC$ and the line $AO$ intersects segment $BC$ at point $T$ . Assume that lines $m$ and $\ell$ passing through point $T$ are perpendicular to $AB$ and $AC$ respectively. If $E$ is the point of intersection of $m$ and $OB$ and $F$ is the point of intersection of $\ell$ and $OC$ , prove that $BE = CF$ . (Oleksii Karliuchenko)

gnoka

18.12.2023 10:31

Suppose M,N to be midpoint of AB,AC $\dfrac{AM}{AD}=\dfrac{AO}{AT}=\dfrac{AN}{AE} \Rightarrow DE //MN$ but MN //BC so $\dfrac{BE}{OB}=\dfrac{BD}{BM}=\dfrac{CE}{CN}=\dfrac{CF}{OC}$ $\therefore BE=CF.$

matinyousefi

31.03.2024 00:27

One could also take $A'$ the reflection of $A$ about $O$ and note that $A'C\|TF$ and $A'B\|TE$ and so $\frac{BE}{BO}=\frac{A'T}{A'O}=\frac{CF}{CO}$ which yields $BE=CF$ .