## Properties and characteristics of the tangent Pythagorean Theorem (direct and inverse). Auxiliary similar triangles

In triangle $A B C$, angle $C$ is a right angle, $A C: A B=3: 5$. A circle with its center on the extension of the leg $A C$ beyond point $C$ touches the extension of the hypotenuse $A B$ beyond point $B$ and intersects the leg $B C$ at point $P$, such that $B P: P C=1: 4$. Find the ratio of the radius of the circle to the leg $B C$.
Apply the tangent-secant theorem.

## Solution

Let $A C=3 x, A B=5 x$. Then $B C=4 x, B P=4 x / 5, P C=16 x / 5$. 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 extension of the hypotenuse $A B$. Then $B M^{2}=B P \cdot B Q=144 x^{2} / 25, A M=5 x+12 x / 5=37 x / 5$. From the similarity of triangles $A O M$ and $A B C$, it follows that $O M: B C=A M: A C=37: 15$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a7c46f4bbe36942e4ae1g-06.jpg?height=669&width=626&top_left_y=0&top_left_x=722)

## Answer

$37: 15$.