Circle & Its Tangent
Tangent
The blue line in the figure above is called the “tangent to the circle c”. Another way of saying it is that the blue line is “tangential” to the circle c. (Pronounced “tangenshull”).
The line barely touches the circle at a single point. If the line were closer to the center of the circle, it would cut the circle in two places and would then be called a secant. In fact, you can think of the tangent as the limit case of a secant. As the secant line moves away from the center of the circle, the two points where it cuts the circle eventually merge into one and the line is then the tangent to the circle.
As can be seen in the figure above, the tangent line is always at right angles to the radius at the point of contact.
Tangents to two circles
Given two circles, there are lines that are tangents to both of them at the same time.
If the circles are separate (do not intersect), there are four possible common tangents:
Two external tangents  Two internal tangents 
If the two circles touch at just one point, there are three possible tangent lines that are common to both:
If the two circles touch at just one point, with one inside the other, there is just one line that is a tangent to both:
If the circles overlap – i.e. intersect at two points, there are two tangents that are common to both:
If the circles lie one inside the other, there are no tangents that are common to both. A tangent to the inner circle would be a secant of the outer circle.
Other definitions
 In trigonometry, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
 In calculus, a line is a tangent to a curve if, at the single point of contact, it has the same slope as the curve.
Tangents through an external point
This page shows how to draw the two possible tangents to a given circle through an external point with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.
Proof
This is the same drawing as the last step in the above animation with lines OJ and JM added.
Argument  Reason  

1  OM = MP = JM  M was constructed as the midpoint of OP and JM=OM because JM was constructed with compass width set from MO 
2  JMO is an isosceles triangle  JM=OM from (1) 
3  ∠JMO = 180–2(∠OJM)  Interior angles of a triangle add to 180°. Base angles of isosceles triangles are equal. 
4  JMP is an isosceles triangle  JM=MP from (1) 
5  ∠JMP = 180–2(∠MJP)  Interior angles of a triangle add to 180°. Base angles of isosceles triangles are equal. 
6  ∠JMP + ∠JMO = 180  ∠JMP and ∠JMO form a linear pair 
7  ∠OJP is a right angle  Substituting (3) and (5) into (6):
(180–2∠MJP) + (180–2∠OJM) = 180 Remove parentheses and subtract 360 from both sides: –2∠MJP –2∠OJM = –180 Divide through by –2:: ∠MJP + ∠OJM = 90 
8  JP is a tangent to circle O and passes through P  JP is a tangent to O because it touches the circle at J and is at right angles to a radius at the contact point. 
p  KP is a tangent to circle O and passes through P  As above but using point K instead of J 
Tangent to a circle at a point
This shows how to construct the tangent to a circle at a given point on the circle with compass and straightedge or ruler. It works by using the fact that a tangent to a circle is perpendicular to the radius at the point of contact. It first creates a radius of the circle, then constructs a perpendicular to the radius at the given point.
Printable stepbystep instructions
The above animation is available as a printable stepbystep instruction sheet, which can be used for making handouts or when a computer is not available.
Proof
The image below is the final drawing above.
Argument  Reason  

1  Line segment OR is a radius of the circle O.  It is a line from the center to the given point P on the circle. 
2  SP is perpendicular to OR  By construction, SP is the perpendicular to OR at P. 
3  SP is the tangent to O at the point P  The tangent line is at right angles to the radius at the point of contact. 
External tangents to two given circles
This page shows how to draw one of the two possible external tangents common to two given circles with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.
How it works
The figure below is the final construction with the line PJ added.
The construction has three main steps:
 The circle OJS is constructed so its radius is the difference between the radii of the two given circles. This means that JL = FP.
 We construct the tangent PJ from the point P to the circle OJS. This is done using the method described in Tangents through an external point.
 The desired tangent FL is parallel to PJ and offset from it by JL. Since PJLF is a rectangle, we need the best way to construct this rectangle. The method used here is to construct PF parallel to OL using the “angle copy” method as shown in Constructing a parallel through a point
As shown below, there are two such tangents, the other one is constructed the same way but on the bottom half of the circles.
Proof
This is the same drawing as the last step in the above animation with line PJ added.
Argument  Reason  

1  PJ is a tangent to the inner circle O at J.  By construction. 
2  FP is parallel to LJ  By construction. 
3  FP = LJ  QS was set from the radius of circle P in construction steps 2 and 3. 
4  FPJL is a rectangle 

5  ∠FLJ = ∠LFP = 90°  Interior angles of rectangles are 90° (4) 
6  FL is a tangent to circle O and P  Touches the circle at one place (F and L), and is at right angles to the radius at the point of contact 
Tangent (tan) function – Trigonometry
The tangent function, along with sine and cosine, is one of the three most common trigonometric functions. In any right triangle, the tangent of an angle is the length of the opposite side (O) divided by the length of the adjacent side (A). In a formula, it is written simply as ‘tan’.
As an example, let’s say we want to find the tangent of angle C in the figure above (click ‘reset’ first). From the formula above we know that the tangent of an angle is the opposite side divided by the adjacent side. The opposite side is AB and has a length of 15. The adjacent side is BC with a length of 26. So we can write
This division on the calculator comes out to 0.577. So we can say “The tangent of C is 0.5776 ” or
Example – using tangent to find a side length
If we look at the general definition –
we see that there are three variables: the measure of the angle x, and the lengths of the two sides (Opposite and Adjacent). So if we have any two of them, we can find the third.
In the figure above, click ‘reset’. Imagine we didn’t know the length of the side BC. We know that the tangent of A (60°) is the opposite side (26) divided by the adjacent side AB – the one we are trying to find.
From our calculator we find that tan 60° is 1.733, so we can write
Transposing:
which comes out to 26, which matches the figure above.
The inverse tangent function – arctan
For every trigonometry function such as tan, there is an inverse function that works in reverse. These inverse functions have the same name but with ‘arc’ in front. So the inverse of tan is arctan etc.
When we see “arctan A”, we interpret it as “the angle whose tangent is A”
tan 60 = 1.733  Means: The tangent of 60 degrees is 1.733 
arctan 1.733 = 60  Means: The angle whose tangent is 1.733 is 60 degrees. 