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Prove the statements about chords

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
3,966
Hey!! :eek:

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? (Wondering)

    $$$$
  2. Do you have an idea for this statement? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,632
Leiden
Hey!! :eek:

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? (Nerd)

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. (Worried)

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? (Worried)

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? (Wondering)