4. Given a cube. $A, B$ and $C$ are the midpoints of its edges (see figure). What is the angle $ABC$?

Answer. $120^{\circ}$.
Solution. 1st method. Draw diagonals $D E \| B C$ and $E F \| A B$ and let $K$ be the point on the extension of diagonal $D E$ beyond point $E$ (see figure). Then $\angle A B C = \angle F E K$. But triangle $D E F$ is equilateral, so

![](https://cdn.mathpix.com/cropped/2024_05_06_33af18a859d624d64782g-1.jpg?height=349&width=438&top_left_y=2327&top_left_x=1503)
$\angle D E F = 60^{\circ}$, and therefore, $\angle F E K = 120^{\circ}$.

2nd method. Introduce a coordinate system with the origin at point $O$, axes $O x, O y$, and $O z$ directed along vectors $\overrightarrow{\mathrm{OM}}, \overrightarrow{\mathrm{OF}}$, and $\overrightarrow{\mathrm{OD}}$ respectively, and let the edge of the cube be 2. Then $A(0 ; 0 ; 1), B(1 ; 0 ; 2)$, $C(2 ; 1 ; 2)$. Therefore, $\overrightarrow{\mathrm{BA}}(-1 ; 0 ;-1), |\overrightarrow{\mathrm{BA}}| = \sqrt{2}, \overrightarrow{\mathrm{BC}}(1 ; 1 ; 0), |\overrightarrow{\mathrm{BC}}| = \sqrt{2}$. Now we find the scalar product of vectors $\overrightarrow{\mathrm{BA}}$ and $\overrightarrow{\mathrm{BC}}$ in two ways:

$\overrightarrow{\mathrm{BA}} \cdot \overrightarrow{\mathrm{BC}} = (-1) \cdot 1 + 0 \cdot 1 + (-1) \cdot 0 = -1$ and $\overrightarrow{\mathrm{BA}} \cdot \overrightarrow{\mathrm{BC}} = \sqrt{2} \cdot \sqrt{2} \cos A B C$. From these two equations, we get that $\cos A B C = -0.5$, i.e., the angle $A B C$ is $120^{\circ}$.

3rd method. Let the edge of the cube be 1. Then, by the Pythagorean theorem, $A B = B C = \frac{\sqrt{2}}{2} ; D C = \frac{\sqrt{5}}{2}$ and $A C = \sqrt{\mathrm{AD}^{2} + \mathrm{DC}^{2}} = \sqrt{\frac{3}{2}}$. Now, using the cosine rule in triangle $A B C$, we find that $\cos A B C = -0.5$.

Grading criteria.

- Correct answer with all justifications - 7 points.
- Correct solution process, but incorrect answer due to an arithmetic error - 5 points.
- Answer of $60^{\circ} - 4$ points.
- Only the answer (including the correct one) - 0 points.