# Prove the statements about chords

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!

Let $C$ be a circle. I want to prove the following:
1. If a chord of the circle changes the position keeping its length, then the midpoint is contained in a circle $C'$ concentric to $C$.
2. If two chords of $C$ have their midpoints on a concentric circle $C'$, then they are equal.

I have done the following:
1. 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?


2. Do you have an idea for this statement?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hey!!

Let $C$ be a circle. I want to prove the following:
1. If a chord of the circle changes the position keeping its length, then the midpoint is contained in a circle $C'$ concentric to $C$.
2. If two chords of $C$ have their midpoints on a concentric circle $C'$, then they are equal.
I have done the following:
1. 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 $90^{\circ}$.
Hey mathmari !!

It is true, but how do we know?
Do you have a proposition for it?

So, the circumcircle of $ODP$ has diameter OP, which is fixed for all chords.
That does not look correct.
The circumcircle of $ODP$ is a different circle that does not have diameter OP.

Therefore, all the midpoints lie on a circle with diameter $OP$, and so all the midpoints lie on a circle with center $O$.
The midpoints are on a circle with diameter $OD$ instead of $OP$, aren't they?

2. If two chords of $C$ have their midpoints on a concentric circle $C'$, then they are equal.

Do you have an idea for this statement?
Suppose we pick a point $D$ on the inner circle.
And draw a line tangential to the circle, which intersects the outer circle at $P$ and $P'$.

What can we say about the triangles $ODP$ and $ODP'$?
Are they congruent?