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Two tangents drawn from one point

Auxiliary similar triangles Properties and characteristics of isosceles triangles.

Inscribed and circumscribed circles

Pythagorean theorem (direct and inverse)

In an isosceles triangle \(ABC (AB = BC)\), a circle is inscribed. A line parallel to side \(BC\) and tangent to the circle intersects side \(AB\) at a point \(N\) such that \(AN = \frac{3}{8} AB\). Find the radius of the circle if the area of triangle \(ABC\) is 12.

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Let the tangent intersect the base $A C$ at point $M$, and the circle touches the lines $M N, A B$ and $A C$ at points $D$, and $H$ respectively. Denote

$B C=A B=a, A H=C H=b$. Then $A E=A H=b, M D=M H, N D=N E, A E=A N+N E=A N+N D, A H=A M+$ $M H=A M+M D, A E=A H$.

If $p^{\prime}$ and $p-$ are the semiperimeters of similar triangles $A N M$ and $A B C$, then $p^{\prime}=A E=A H=b, p=A B+A H=a+b$, $p^{\prime}: p=A N: A B_{1}=3: 8$, or

$b / a+b=3 / 8$. Hence, $b=3 a / 5$. Therefore, $B H^{2}=A B^{2}-A H^{2}=a^{2}-\left({ }^{3 a} / 5\right)^{2}=\left({ }^{4 a} / 5\right)^{2}$, and since $S_{A B C}=A H \cdot B H$, then $12=$ $b \cdot{ }^{4 a} / 5={ }^{3 a} / 5 \cdot{ }^{4 a} / 5$, from which $a=5, b=3$. Let $r$ be the radius of the inscribed circle of triangle $A B C$. Then $r=$ $S_{A B C / p}=12 / a+b=3 / 2$

![](https://cdn.mathpix.com/cropped/2024_05_06_6409e68af5a1ba5e989dg-17.jpg?height=709&width=1003&top_left_y=0&top_left_x=573)

## Answer

$3 / 2$.

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