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- #1

- Apr 14, 2013

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Let $C$ be a circle. I want to prove the following:

- If a chord of the circle changes the position keeping its length, then the midpoint is contained in a circle $C'$ concentric to $C$.
- If two chords of $C$ have their midpoints on a concentric circle $C'$, then they are equal.

I have done the following:

- Let $O$ be the centre of $C$.

Let $D$ be the midpoint of a chord passing through a point $P$.

Since $D$ is the midpoint of the chord, it holds that the angle of $ODP$ is equal to $90^{\circ}$.

So, the circumcircle of $ODP$ has diameter OP, which is fixed for all chords.

Therefore, all the midpoints lie on a circle with diameter $OP$, and so all the midpoints lie on a circle with center $O$.

Is that correct?

$$$$

- Do you have an idea for this statement?