[

[ Tangent Circles $]$

Two circles touch each other internally. A line passing through the center of the smaller circle intersects the larger circle at points $A$ and $D$, and the smaller circle at points $B$ and $C$. Find the ratio of the radii of the circles if $A B: B C: C D=2: 4: 3$.

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

## Solution

Let $M$ be the point of tangency of the circles, $r$ and $R (r < R)$ be their radii, and $O$ be the center of the smaller one. Denote: $AB = 2x, BC = 4x, CD = 3x$. Then

$$
r = OB = OC = 2x, DO \cdot OA = MO(2R - MO), \text{ or } 5x \cdot 4x = 2x(2R - 2x).
$$

Therefore,

$$
5x = R - x, R = 6x, \frac{R}{r} = \frac{6x}{2x} = 3
$$

Let $M$ be the point of tangency of the circles, $r$ and $R (r < R)$ be their radii, and $O$ be the center of the smaller one. Denote: $AB = 2x, BC = 4x, CD = 3x$. Then

$$
r = OB = OC = 2x, DO \cdot OA = MO(2R - MO), \text{ or } 5x \cdot 4x = 2x(2R - 2x).
$$

Therefore,

$$
5x = R - x, R = 6x, \frac{R}{r} = \frac{6x}{2x} = 3
$$

Let $M$ be the point of tangency of the circles, $r$ and $R (r < R)$ be their radii, and $O$ be the center of the smaller one. Denote: $AB = 2x, BC = 4x, CD = 3x$. Then

$$
r = OB = OC = 2x, DO \cdot OA = MO(2R - MO), \text{ or } 5x \cdot 4x = 2x(2R - 2x)
$$

Therefore,

$$
5x = R - x, R = 6x, \frac{R}{r} = \frac{6x}{2x} = 3
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_ae1b0d3bbbf9aa5c38e8g-55.jpg?height=460&width=498&top_left_y=-1&top_left_x=783)

## Answer

3.