[Theorem of the tangent and secant; the product of the entire secant and its external part
Properties and signs of the tangent
Pythagorean Theorem (direct and inverse).
Auxiliary similar triangles

In triangle $ABC$, angle $C$ is a right angle, $AC: AB = 4: 5$. A circle with its center on the leg $AC$ touches the hypotenuse $AB$ and intersects the leg $BC$ at point $P$, such that

$BP: PC = 2: 3$. Find the ratio of the radius of the circle to the leg $BC$.

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Apply the tangent-secant theorem.

## Solution

Let $A C=4 x, A B=5 x$. Then $B C=3 x, B P=6 x / 5, P C=9 x / 5$.

Let $Q$ be the second intersection point of line $B C$ with the given circle, $O$ be the center of this circle, and $M$ be the point of tangency with the hypotenuse. Then

$B M^{2}=B P(B P+P C+C Q)=144 x^{2} / 25$. Therefore, $A M=5 x-12 x / 5=13 x / 5$. From the similarity of triangles $A O M$ and $A B C$, it follows that $O M: B C=13: 20$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a7c46f4bbe36942e4ae1g-05.jpg?height=528&width=548&top_left_y=868&top_left_x=755)

## Answer

$13: 20$.