Marinada '24

[lang=hr] U trokutu $ABC$ vrijedi $AB=360$, $BC=507$, $CA=780$. Neka je $M$ polovište stranice $\overline{CA}$, $D$ točka na $\overline{CA}$ tako da je $\overline{BD}$ simetrala $\angle ABC$. Neka je $E$ točka na $\overline{BC}$ takva da je $DE \perp BD$. Neka je $F=DE \cap BM$. Nađi $\frac{DF}{EF}$. Odgovor zaokruži na $3$ decimale. [/lang] [lang=en] In $\triangle ABC$, $AB = 360$, $BC = 507$, and $CA = 780$. Let $M$ be the midpoint of $\overline{CA}$, and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC$. Let $E$ be the point on $\overline{BC}$ such that $\overline{DE} \perp \overline{BD}$. Suppose that $\overline{DE}$ meets $\overline{BM}$ at $F$. Find $\frac{DF}{EF}$ and round your solution to $3$ decimal places. [/lang]