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.

Given: A tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

To do: To 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 figure, C is the midpoint of the minor arc PQ, O is the center of the circle and AB is tangent to the circle through point C.

We have to show the tangent drawn at the midpoint of the arc PQ of a circle is parallel to the chord joining the end points of the arc PQ.

We will show $PQ \parallel AB$.

It is given that C is the midpoint point of the arc PQ.

So, $arc\ PC = arc\ CQ$.

$PC = CQ$

This shows that 

$\vartriangle PQC$ is an isosceles triangle.

Thus, the perpendicular bisector of the side PQ of $\vartriangle PQC$ passes through vertex C.

The perpendicular bisector of a chord passes through the center of the circle.

So the perpendicular bisector of PQ passes through the center O of the circle.

Thus perpendicular bisector of PQ passes through the points O and C.

therefore, PQ$\perp $OC

AB is the tangent to the circle through the point C on the circle.

Therefore, AB $\perp $OC

The chord PQ and the tangent PQ of the circle are perpendicular to the same line OC.

$PQ\parallel AB$.