[  The product of the lengths of the segments of chords and the lengths of the segments of secants [Theorem on the lengths of the tangent and secant; the product of the entire secant and its external part

In triangle $A B C$, it is known that $A B=B C, A C=4 \sqrt{3}$, and the radius of the inscribed circle is 3. Line $A E$ intersects the height $B D$ at point $E$, and the inscribed circle at points $M$ and $N$ (with $M$ lying between $A$ and $E$), and $E D=2$. Find $E N$.

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Denote $M E=x, E N=y$. By applying the theorem of the equality of the products of segments of intersecting chords of a circle and the theorem of the tangent and secant, form a system of equations in terms of $x$ and $y$.

## Solution

Let the incircle of triangle $A B C$ intersect the height $B D$ at point $K$ for the second time. Denote $M E=x$, $E N=y$. According to the theorem of the equality of the products of segments of intersecting chords of a circle, $M E \cdot E N=D E \cdot E K$, or $x y=2 \cdot 4=8$.

From the right triangle $A E D$, we find that

$$
A E=\sqrt{A D^{2}+D E^{2}}=\sqrt{(2 \sqrt{3})^{2}+2^{2}}=4
$$

By the theorem of the tangent and secant, $A M \cdot A N=A D^{2}$, or $(4-x)(4+y)=12$. From the system

$$
\left\{\begin{array}{l}
x y=8 \\
(4-x)(4+y)=12
\end{array}\right.
$$

we find that $y=\frac{1+\sqrt{33}}{2}$.

## Answer

$\frac{1+\sqrt{33}}{2}$.