Three segments with endpoints on the sides of a triangle, parallel to its sides, pass through one point and have the same length $x$.

Find $x$, if the sides of the triangle are $a, b, c$.

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Consider similar triangles.

## Solution

Let's denote the vertices of all triangles as shown in the figure. By the condition, \( M L = E N = K F = x \). Let \( A B = c, A C = b, B C = a \). Since

\( L K = A B - B L - A K = A B - E P - N P = A B - E N = c - x \), the similarity coefficient of triangles \( L P K \) and \( B C A \) is \( \frac{c - x}{c} \). Similarly, we find that the similarity coefficient of triangles \( E F P \) and \( B C A \) is \( \frac{a - x}{a} \). Therefore, \( A N = P K = A C \cdot \frac{c - x}{c} = b \cdot \frac{c - x}{c} \), and \( C M = P F = A C \cdot \frac{a - x}{a} = b \cdot \frac{a - x}{a} \).

Thus, \( b = A C = C M + M N + N A = b \cdot \frac{a - x}{a} + b - x + b \cdot \frac{c - x}{c} \).

From the obtained equation, we find that \( x = \frac{2 a b c}{a b + a c + b c} \).

![](https://cdn.mathpix.com/cropped/2024_05_06_9bfea73962495faf8ed9g-12.jpg?height=315&width=666&top_left_y=1849&top_left_x=702)

## Answer

\(\frac{2 a b c}{a b + a c + b c}\)