Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.
In the given figure, C is the mid point of the minor arc AB of the circle with centre O.
PQ is the tangent to the given circle through point C.
To Prove: Tangent drawn at the mid point of the arc
i.e AB || PQ.
Proof: C is the mid point of the minor arc AB
⇒ minor arc AC = minor arc BC
⇒ AC = BC
Thus, ∆ABC is an iscosceles triangle.
Therefore, the perpendicular bisector of side AB of ∆ABC passes through the vertex C.
We know that the perpendicular bisector of a chord passes through the centre of the circle.
Since AB is a chord of the circle so, the perpendicular bisector of AB passes through the centre O.
Thus, it is clear that the perpendicular bisector of AB passes through the points O and C.
Now, PQ is the tangent to the circle through the point C on the circle.
Let us draw a circle in which AMB is an arc and M is the mid-point of the arc AMB.
Joined AM and MB. Also TT' is a tangent at point M on the circle.
To Prove: AB || TT'
Proof: As M is the mid point of Arc AMB
Arc AM = Arc MB
AM = MB .......[As equal chords cuts equal arcs]
∠ABM = ∠BAM .......[Angles opposite to equal sides are equal] 
Now, ∠BMT' = ∠BAM ......[Angle between tangent and the chord equals angle made by the chord in alternate segment] 
From  and 
∠ABM = ∠BMT'
So, AB || TT' [two lines are parallel if the interior alternate angles are equal]