[ The product of the lengths of the segments of chords and the lengths of the segments of secants ]

From point $A$, two rays are drawn intersecting a given circle: one at points $B$ and $C$, and the other at points $D$ and $E$. It is known that $A B=7, B C=7, A D=10$. Find $D E$.

#
Apply the corollary of the tangent-secant theorem.

## Solution

It is clear that point $B$ is located between points $A$ and $C$. Suppose that point $D$ is located between points $A$ and $E$. Then, by the corollary of the tangent-secant theorem,

$$
A B \cdot A C = A D \cdot A E, \text{ or } 14 \cdot 7 = 10(10 + D E).
$$

From this, we find that $D E = -\frac{1}{5}$, which is impossible. Therefore, point $E$ is located between $A$ and $D$. Then,

$$
A B \cdot A C = A D \cdot A E, \text{ or } 14 \cdot 7 = 10(10 - D E).
$$

From this, we find that $D E = \frac{1}{5}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b0b755a355dbd05a96cdg-26.jpg?height=400&width=606&top_left_y=1352&top_left_x=726)

Answer

$\frac{1}{5}$