In the figure below, $ABCD$ is a rectangle and $E$ is the midpoint of $AD$. The segment $FG$ passes through the midpoint $M$ of $CE$. Determine the ratio between the lengths of $GM$ and $MF$.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-16.jpg?height=351&width=580&top_left_y=1115&top_left_x=652)
Solution

From point $M$, draw the line segment $P Q$ perpendicular to sides $A B$ and $C D$ of rectangle $A B C D$ as shown in the figure below. Since $M$ is the midpoint of $C E$, we can conclude that $P M$ is the midline relative to side $D E$ of triangle $E C D$. Thus, if $D E=E A=x, P M=$ $D E / 2=x / 2$. Since $E$ is the midpoint of $D A$, we have $P Q=D A=2 x$. Consequently, $M Q=2 x-P M=3 x / 2$. Triangles $\triangle P M G$ and $\triangle M F Q$ are similar, as they have the same angles. Therefore,

$$
\frac{G M}{M F}=\frac{P M}{M Q}=\frac{x / 2}{3 x / 2}=\frac{1}{3}
$$

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-16.jpg?height=454&width=737&top_left_y=2240&top_left_x=568)