Marinada '23

[lang = hr] Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much? Write your answer in the form "XY" where X is the first letter of the winner's name and Y is a difference rounded to $2$ decimal places. [/lang] [lang = en] Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{ BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much? Write your answer in the form "XY" where X is the first letter of the winner's name and Y is a difference rounded to $2$ decimal places. [/lang]