Problem

Source: Stars of Mathematics 2023 P3 (junior level)

Tags: geometry, Fixed point



Let $ABC$ be an acute triangle, with $AB<AC{}$ and let $D$ be a variable point on the side $AB{}$. The parallel to $D{}$ through $BC{}$ crosses $AC{}$ at $E{}$. The perpendicular bisector of $DE{}$ crosses $BC{}$ at $F{}$. The circles $(BDF)$ and $(CEF)$ cross again at $K{}$. Prove that the line $FK{}$ passes through a fixed point. Proposed by Ana Boiangiu