
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
(See Constructing the perpendicular bisector of a line segment for method and proof)
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.
(see Tangent to a circle.)

p 
KP is a tangent to circle O and passes through P 
As above but using point K instead of J

 Q.E.D
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.
Try it yourself
Click here for a printable tangents construction worksheet with some problems to try.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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