Tangent

On a Circle
1. Begin with a circle centered on O with a point A on it.

2. Construct the line $$\overset{\longleftrightarrow}{O A}$$.

3. Construct an arbitrary circle centered on A. It intersects $$\overset{\longleftrightarrow}{O A}$$ at points B and C.

4. Construct the perpendicular bisector of $$\overline{B C}$$. This line is tangent to the circle $$\overset{\frown}{O A}$$ at point A.

Off a Circle
1. Begin with a circle centered on O with a point A outside of it.

2. Construct the midpoint of O and A.

3. Construct the circle $$\overset{\frown}{M O}$$. It intersects the other circle at points B and C.

4. Construct the lines $$\overset{\longleftrightarrow}{A B}$$ and $$\overset{\longleftrightarrow}{A C}$$.