## [ Rectangles and Squares. Properties and Characteristics ]  [ Inscribed Quadrilaterals (other). ]

Point $F$ is the midpoint of side $B C$ of square $A B C D$. A perpendicular $A E$ is drawn to segment $D F$. Find the angle $C E F$.

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Let the line $A E$ intersect the side $C D$ of the square at point $M$ (see the figure). Then triangles $A D M$ and $D C F$ are equal (by the leg and acute angle). Therefore, point $M$ is the midpoint of side $C D$. Hence, triangle $C F M$ is a right isosceles triangle, so $\angle C M F=45^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_5ddd4eab4fe1a12edeb9g-34.jpg?height=546&width=554&top_left_y=2237&top_left_x=755)

Since $\angle M E F=\angle M C F=90^{\circ}$, quadrilateral $M C F E$ is cyclic. Therefore, $\angle C E F=\angle C M F=45^{\circ}$.