A line passing through a common point $A$ of two circles intersects these circles again at points $B$ and $C$ respectively. The distance between the projections of the centers of the circles onto this line is 12. Find $BC$, given that point $A$ lies on the segment $BC$.
The diameter perpendicular to a chord bisects it.

## Solution

Let $M$ and $N$ be the projections of the centers $O_{1}$ and $O_{2}$ of the given circles onto the line $B C$ (with $M$ on $A B$ and $N$ on $A C$). Since the diameter perpendicular to a chord bisects it, points $M$ and $N$ are the midpoints of segments $A B$ and $C D$.

Therefore, $B C = A B + A C = 2 A M + 2 A N = 2 M N = 24$.

![](https://cdn.mathpix.com/cropped/2024_05_06_837b6360ec703a44a031g-28.jpg?height=317&width=491&top_left_y=150&top_left_x=795)

## Answer

24.