$[$ Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part

From point $A$, two rays intersect a given circle: one - at points $B$ and $C$, 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 tangent-secant theorem. Consider two cases.

## 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

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

From this, we find that $D E=-0.2$, which is impossible. Therefore, point $E$ is located between $A$ and $D$. Then $A B \cdot A C=A D \cdot A E$, or $14 \cdot 7=10(10-D E)$.

From this, we find that $D E=0.2$.

## Answer

0.2.