[ Properties and signs of tangents ] [ Rectangles and squares. Properties and signs ]

Two circles are given. Their common internal tangents are perpendicular to each other. The chords connecting the points of tangency are 3 and 5. Find the distance between the centers of the circles.
For each of the circles, the quadrilateral formed by the tangents and radii drawn to the points of tangency is a square.

## Solution

For each of the circles, the quadrilateral formed by the tangents and radii drawn to the points of tangency is a square. The segment connecting the point of intersection of the tangents with the center of the circle is the diagonal of the square. The sum of these segments is the distance between the centers.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-32.jpg?height=400&width=769&top_left_y=0&top_left_x=650)

## Answer

8.