A fixed circle is touched from the inside by a rolling circle of half the radius. What trajectory does a fixed point $K$ on the rolling circle describe?

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Draw a diameter through the initial point of tangency.

## Solution

Let $O$ be the center of the fixed circle, $K_{0}$ the initial point of tangency, $O_{1}$ the new center of the rolling circle, $M$ the new point of tangency, and $K$ the moving point. Then $\cup M K_{0}=\cup M K$.

Therefore, point $K$ lies on the line $O K_{0}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_5827e7e7a03e1b3b311bg-50.jpg?height=458&width=529&top_left_y=157&top_left_x=770)

## Answer

The diameter of the circle.