2. Let ABCD be a parallelogram and E, F the midpoints of segments [BC], and respectively [CD]. If $\mathrm{AE} \cap \mathrm{BF}=\{\mathrm{G}\}$, and $\mathrm{H} \in(\mathrm{AG}), \mathrm{with}[\mathrm{AH}] \equiv[\mathrm{HG}]$, find $\frac{H G}{H E}$.

(Sorin Furtună, Stelică Pană, Olympiads and Mathematics Competitions V-VIII 2014, Bîrchi Publishing)
Subject 2.

| Details of solution | Associated grading |
| :---: | :---: |
| ..................................................................... $\quad$ - | $1 \mathrm{p}$ |
| - $\quad$ take L and K as the midpoints of sides [AD] and [AB]........................................................................................ | $1 \mathrm{p}$ |
| - $\quad$ BFDK parallelogram $=>$ DK is the support line of the midline for $\Delta \mathrm{ABG} \Rightarrow \mathrm{H} \in \mathrm{DK} . . . . . . . .$. | $1 \mathrm{p}$ |
| ![](https://cdn.mathpix.com/cropped/2024_06_07_e81ea2b65f6740cce1bdg-2.jpg?height=61&width=1558&top_left_y=1438&top_left_x=196) | $1 \mathrm{p}$ |
| - $\quad N E=\frac{F C}{2}=\frac{A B}{4}=$ LM........................................................................... | $1 \mathrm{p}$ |
| - in $\triangle \mathrm{EMH}, \mathrm{GN} \\| \mathrm{MH}$, apply Thales' theorem $\frac{H G}{H E}=\frac{M N}{M E}=\frac{\frac{A B}{2}}{\frac{3 A B}{4}}=\frac{2}{3}$. | 2p |