Draw any circle. Take any point A on it and construct tangent at A without using the centre of the circle.

Answer: Use the converse of the tangent-secant angle theorem: make the angle between the chord and the line equal to the inscribed angle on the chord.

Analysis: By the converse of the tangent-secant angle theorem, if a line through a point on the circle makes with a chord an angle equal to the angle inscribed in the opposite arc, the line is a tangent.

Steps of construction:

1. Draw a circle of any radius and take a point A on it.
2. Draw a chord AQ through A. Take any point R on the circle, different from A and Q (on the major arc). Join RA and RQ.
3. With R as centre draw an arc of any radius cutting RA and RQ at M and N.
4. With the same radius and centre A, draw an arc cutting chord AQ at S (on the side away from R).
5. With S as centre and radius equal to MN, cut the arc of step 4 at T. Draw line AT.
6. Line AT is the required tangent, since \(\angle QAT = \angle QRA\).