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$[$ Diameter, main properties $]$

Find the geometric locus of the midpoints of all chords passing through a given point of a circle.

#
If $M$ is the midpoint of the chord $A B$ of a circle with center at point $O$ (where $A B$ is not a diameter), then $\angle A M O$ is a right angle.

## Solution

First method.

Let $O$ be the center of the given circle, $A$ a point on the circle, and $M$ the midpoint of the chord $A B$ (which is not a diameter). Then the segment $O A$ is seen from point $M$ at a right angle, since a diameter passing through the midpoint of a chord (which is not a diameter) is perpendicular to it. Therefore, the midpoint of any chord $A B$ (including point $O$) lies on the circle with diameter $A O$.

Conversely, any point on this circle (except point $A$) is the midpoint of some chord $A B$, since a diameter perpendicular to a chord bisects it.

## Second method.

The desired geometric locus of points is a circle (excluding point $A$), homothetic to the given one with a coefficient of $\frac{1}{2}$ and center at point $A$.

![](https://cdn.mathpix.com/cropped/2024_05_06_95d66af6ee6f6b4b6717g-47.jpg?height=443&width=506&top_left_y=1368&top_left_x=387)

Fig. 1.

![](https://cdn.mathpix.com/cropped/2024_05_06_95d66af6ee6f6b4b6717g-47.jpg?height=437&width=621&top_left_y=1371&top_left_x=1050)

## Answer

A circle (excluding a point), the radius of which is half the radius of the given circle.