prove that the line joining the mid-point of two equal chords of a circle subtends equal angles with the chord.

Explanation

Let `AB` and `CD` be the equal chords.
Let `O` be the centre of circle.
Let `M` and `N` be the mid-points of the chords.
`therefore MB=AM`
`CN=DN`
Equal chords are equidistant from the centre.
`:.OM=ON`
In `/_\OAB` and `/_\OCD`
`OA=OD` (Radius of circle)
`AB=CD` (Given)
`OB=OC` (Radius of circle)
`/_\OAB~=/_\OCD` [By `SAS` congruency]
`/_AOB=/_COD` [By `CPCT`]
`therefore` The line joining the mid-point of two equal chords of a circle subtends equal angles with the chord.
Hence Proved.