The non-parallel sides of the trapezoid are extended to intersect each other, and a line parallel to the bases of the trapezoid is drawn through the obtained point. Find the length of the segment of this line bounded by the extensions of the diagonals, if the lengths of the bases of the trapezoid are $a$ and $b$.

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Consider pairs of similar triangles.

## Solution

Let $A D=b$ and $B C=a(b>a)$ be the bases of the trapezoid $A B C D, P$ be the intersection point of lines $A B$ and $C D, M$ and $N$ be the intersection points of lines $D B$ and $A C$ with the line passing through point $P$ parallel to the bases of the trapezoid. From the similarity of triangles $B P C$ and $A P D$, it follows that $P C: P D=B C: A D=a: b$, and from the similarity of triangles $P C N$ and $D C A-P N: A D=P C: C D=a:(b-a)$.

From this, we find that $P N=A D \cdot a / b-a = a b / b-a$.

Similarly, $M P=a b / b-a$. Therefore, $M N=P N+M P=2 a b / b-a$.

If $a>b$, then $M N=2 a b / b-a$.

![](https://cdn.mathpix.com/cropped/2024_05_06_96d74c2697cd905bd948g-18.jpg?height=326&width=1146&top_left_y=606&top_left_x=456)

Answer

$2 a b /|a-b|$