$_{[}^{[1}$  [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part The ratio in which the bisector divides the side Properties and criteria of an isosceles triangle.

In an isosceles triangle $A B C$, the bisectors $A D, B E, C F$ are drawn.

Find $B C$, if it is known that $A B=A C=1$, and vertex $A$ lies on the circle passing through points $D, E$ and $F$.

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Apply the property of the bisector of a triangle and the tangent-secant theorem.

## Solution

Let $B C=x$. By the property of the bisector of a triangle, $C E: A E=x: 1$, hence $C E=x / 1+x$. Since $C D^{2}=$ $C E \cdot A C$, then $x^{2} / 4=x / 1+x$. From this equation, we find that $x=\frac{\sqrt{17}-1}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-30.jpg?height=326&width=604&top_left_y=1921&top_left_x=727)

## Answer

$\frac{\sqrt{17}-1}{2}$