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The text in the image states the following problem in Russian:

"In an isosceles triangle ABC with base AC, the medians drawn to the lateral sides are perpendicular. Find the cosine of angle ABC."

Let's analyze and solve the problem:

1. **Given**: Isosceles triangle \(ABC\), where \(AC\) is the base, and the medians drawn from vertex \(A\) and vertex \(C\) to the opposite sides (i.e., to sides \(AB\) and \(BC\)) are perpendicular.

2. **Objective**: Find \(\cos(\angle ABC)\).

### Solution:
Let us define the triangle's points and elements:

- Triangle \(ABC\) is isosceles, so \(AB = BC\).
- Let the median from vertex \(A\) to side \(BC\) be \(AM\), and the median from vertex \(C\) to side \(AB\) be \(CN\). According to the problem, \(AM \perp CN\).

In an isosceles triangle, the medians from the vertices of the equal sides are also angle bisectors and altitudes, making them perpendicular. This geometric constraint leads to a key trigonometric relationship.

Since the triangle has symmetry and the medians are perpendicular, this configuration suggests that \(\angle ABC = 90^\circ\). Therefore, the cosine of a right angle is:

\[
\cos(\angle ABC) = \cos(90^\circ) = 0.
\]

Thus, the cosine of angle \(ABC\) is 0.

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Would you like further details on any step or a different explanation?

Here are 5 related questions to explore further:

1. How do you calculate the cosine of an angle in a right triangle?
2. Why are the medians in an isosceles triangle perpendicular in this scenario?
3. What properties define the medians in any triangle?
4. How does symmetry in isosceles triangles affect angles and side lengths?
5. How would the problem change if the medians were not perpendicular?

**Tip**: In any triangle, the median divides the opposite side into two equal parts. This property can help in symmetry-based problems like this one!

In an isosceles triangle ABC with base AC, the medians drawn to the lateral sides are perpendicular. Find the cosine of angle ABC.

Learn how to solve for the cosine of angle ABC in an isosceles triangle where the medians to the lateral sides are perpendicular. This problem explores geometric concepts and trigonometric properties.

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