In a circle, the line segment joining the midpoints of two chords passes through the centre. Show that two chords are parallel
Proof:
\(PQ\) is the diameter of the circle bisecting the chord \(AB\) and \(CD\) at points \(L\) and \(M\) respectively.
Now, we need to prove: \(AB ∥ CD\)
Now, is the mid-point of \(AB\).
By theorem,
Therefore, \(OL⊥\) .
That is, \(∠ALO=\)\(^°\) .....(1)
Also, \(OM⊥ \).
Therefore, \(∠OMD=\)\(^°\) .....(2)
From (1) and (2), \(∠ALO=∠OMD=\)\(^°\)
Since, these are alternate interior angles \(FE\) is
Therefore, \(AB ∥ CD\).
Hence, proved.