[ Tangent Circles [ Equilateral (Equiangular) Triangle ]

In a given circle with a radius of 3, six equal circles are inscribed, each touching the given circle; moreover, each of these six circles touches two adjacent ones. Find the radii of the circles.

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Consider a triangle with vertices at the centers of the external and two adjacent internal circles.

## Solution

Let $r$ be the radius of one of the six internal circles. The triangle with vertices at the centers of the external and two adjacent internal circles is equilateral, so $3-r=2r$. Therefore, $r=1$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b17058527caf5b5d058g-04.jpg?height=686&width=712&top_left_y=1987&top_left_x=679)

## Answer

1. 

Six equal circles touch externally a circle of radius 1 and, in addition, each of these six circles touches two adjacent ones. Find the radii of the circles.

## Hint

Consider a triangle with vertices at the centers of the internal and two adjacent external circles.

## Solution

Let $r$ be the radius of one of the six external circles. The triangle with vertices at the centers of the internal and two adjacent external circles is equilateral. Therefore, $1+r=2r$. Consequently, $r=1$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b17058527caf5b5d058g-05.jpg?height=606&width=698&top_left_y=589&top_left_x=680)

## Answer

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