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Two circles touch each other internally. A line passing through the center of the larger circle intersects it at points $A$ and $D$, and the smaller circle at points $B$ and $C$. Find the ratio of the radii of the circles if $A B: B C: C D=3: 7: 2$.

#
Apply the theorem of the products of segments of intersecting chords.

## Solution

Let $M$ be the point of tangency, $r$ and $R (r < R)$ be the radii of the circles, and $Q$ be the center of the larger one. Denote $A B = 3 x, B C = 7 x, C D = 2 x$. Then

$$
\begin{gathered}
R = \frac{A B + B C + C D}{2} = 6 x, B Q = A Q - A B = R - 3 x = 3 x \\
Q C = R - 2 x = 4 x, M Q = R = 6 x, Q P = 2 r - M Q = 2 r - 6 x
\end{gathered}
$$

(where $M P$ is the diameter of the smaller circle).

By the theorem of the products of segments of intersecting chords

$$
B Q \cdot Q C = M Q \cdot Q P, \text{ or } 3 x \cdot 4 x = 6 x \cdot (2 r - 6 x)
$$

From this equation, we find that $r = 4 x$. Therefore,

$$
\frac{R}{r} = \frac{6 x}{4 x} = \frac{3}{2}
$$

Let $M$ be the point of tangency, $r$ and $R (r < R)$ be the radii of the circles, and $Q$ be the center of the larger one. Denote $A B = 3 x, B C = 7 x, C D = 2 x$. Then

$$
\begin{gathered}
R = \frac{A B + B C + C D}{2} = 6 x, B Q = A Q - A B = R - 3 x = 3 x \\
Q C = R - 2 x = 4 x, M Q = R = 6 x, Q P = 2 r - M Q = 2 r - 6 x
\end{gathered}
$$

(where $M P$ is the diameter of the smaller circle).

By the theorem of the products of segments of intersecting chords

$$
B Q \cdot Q C = M Q \cdot Q P, \text{ or } 3 x \cdot 4 x = 6 x \cdot (2 r - 6 x)
$$

From this equation, we find that $r = 4 x$. Therefore,

$$
\frac{R}{r} = \frac{6 x}{4 x} = \frac{3}{2}
$$

Let $M$ be the point of tangency, $r$ and $R (r < R)$ be the radii of the circles, and $Q$ be the center of the larger one. Denote $A B = 3 x, B C = 7 x, C D = 2 x$. Then

$$
\begin{gathered}
R = \frac{A B + B C + C D}{2} = 6 x, B Q = A Q - A B = R - 3 x = 3 x \\
Q C = R - 2 x = 4 x, M Q = R = 6 x, Q P = 2 r - M Q = 2 r - 6 x
\end{gathered}
$$

By the theorem of the products of segments of intersecting chords

$$
B Q \cdot Q C = M Q \cdot Q P, \text{ or } 3 x \cdot 4 x = 6 x \cdot (2 r - 6 x)
$$

From this equation, we find that $r = 4 x$. Therefore,

$$
\frac{R}{r} = \frac{6 x}{4 x} = \frac{3}{2}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_ae1b0d3bbbf9aa5c38e8g-54.jpg?height=437&width=460&top_left_y=342&top_left_x=799)

## Answer

$3: 2$.