This demonstration shows how to construct the two possible tangents to a given circle through a given external point using only a compass and straight edge. See "Introduction to Constructions" We start with a given circle with center O and and a point P outside the circle. The result is two tangent lines that pass through P.

To construct a tangent at a point on the circle see Tangent to a circle - construction.

Step 1	Draw a straight line between the center O of the given circle and the given point P.
Step 2	Find the midpoint of this line by constructing the line's perpendicular bisector. See Constructing the Perpendicular Bisector of a Line Segment.
Step 3	Place the compass on the midpoint just constructed, and set it's width so that it passes through the center O of the circle..
Step 4	Without changing the width, draw an arc across the circle in the two possible places. These are the contact points J, K for the tangents.
Step 5	Draw the two tangent lines from P through J and K.
Step 6	Done. The two lines just drawn are tangential to the given circle and pass through P.