We start with a point P somewhere on a given circle, with center point O.
If the center is not given, you can use: "Finding the center of a circle with compass and straightedge or ruler",
or
"Finding the center of a circle with any rightangled object".


1. Draw a straight line from the center O, through the given point P and on beyond P. 

In the following steps 2  6 we are constructing the perpendicular to the line OP at a point P.
This is the same procedure as described in Constructing a perpendicular at a point on a line.

2. Put the compasses' point on P and set it to any width less than the distance OP.
Then, on the line just drawn, draw an arc on each side of P. This creates the points Q and R as shown. 

3. Set the compasses on Q and set it to any width greater than the distance QP. 

4. Without changing the compasses' width, draw an arc approximately in the position shown on one side of P. 

5. Without changing the compasses' width,
move the compasses to R and make another arc across the first, creating point S. 

6. Draw a line through P and S. 

7. Done. The line PS just drawn is the tangent to the circle O through point P. 
