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

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

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.