## Median drawn to the hypotenuse  Problem $\underline{110826}$ Inscribed and circumscribed circles [Area of a triangle (using semiperimeter and radius of inscribed or exscribed circle. Pythagorean theorem (direct and inverse).

In a right triangle $A B C$, a median $C D$ is drawn from the vertex of the right angle $C$. Find the distance between the centers of the circles inscribed in triangles $A C D$ and $B C D$, if $B C=4$, and the radius of the circumscribed circle of triangle $A B C$ is $5 / 2$.

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The radius of the circumscribed circle of a right-angled triangle is half the hypotenuse, so $AB=5$. By the Pythagorean theorem, $AC=3$.

Let $O_{1}$ and $O_{2}$ be the centers of the circles, and $R$ and $r$ be the radii of the circles inscribed in triangles $ACD$ and $BCD$ respectively. Then

![](https://cdn.mathpix.com/cropped/2024_05_06_95d66af6ee6f6b4b6717g-19.jpg?height=85&width=2027&top_left_y=2450&top_left_x=10)
$O_{1} O_{2}=\sqrt{D O_{1}^{2}+D O_{2}^{2}}=\frac{5 \sqrt{13}}{12}$

![](https://cdn.mathpix.com/cropped/2024_05_06_95d66af6ee6f6b4b6717g-20.jpg?height=669&width=514&top_left_y=0&top_left_x=769)

## Answer

$\frac{5 \sqrt{13}}{12}$.

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