
How to construct a line tangent to a given circle, through a given point on or outside that circle?

1. 
 Start with the given circle and point.
If the given point lies inside the circle, there is no such line.
First case: the given point lies on the circle.


2. 
 Draw the ray of the given circle from its center to the given point.


3. 
 Construct the line perpendicular to ray 2, passing through the given point.
This is the line to be constructed: it is passing though the given point and is tangent to the given circle.


4. 
 Case 2: the given point lies outside the given circle.


5. 
 Construct the "twopoints" circle (defined by the two endpoints of a centerline) between the given point and the center of the given circle.


6. 
 Draw two lines, from the given point to the two intersections of the given circle and circle 5.


7. 
 Lines 6 are the lines to be constructed. They pass through the given point, and are tangent to the given circle.
Notice, that there are two such lines!

