[ [MT - circle or arc of a circle] [Inscribed angle subtended by a diameter]

Find the set of midpoints of chords passing through a given point $A$ inside the circle.
Any midpoint of a chord, a given point, and the center of the circle form a right triangle.

## Solution

Let $A$ be the given point, $O$ be the center of the given circle $\Omega$. Consider some chord $BC$ passing through point $A$. Let $P$ be the midpoint of this chord. Point $O$ lies on the perpendicular bisector of segment $BC$, so $OP \perp BC$. Therefore, point $P$ lies on the circle $\omega$ constructed with $OA$ as its diameter. Conversely, if we take any point $P$ on circle $\omega$, it is easy to see that the line $AP$ intersects circle $\Omega$ at a chord for which $P$ is the midpoint.

## Answer

The circle constructed with segment $OA$ as its diameter ($O$ being the center of the original circle).