Remember -- use your compass
and straightedge only!
Tangent to Circle from Point OUTSIDE Circle (G.C.4)+
This construction requires a bit more work. Picture in your mind what these tangents to a circle will look like. The diagram at the right shows two tangents from point P. You need not draw the two tangents, but the two of them may remind you of how this construction will look, as the construction creates two possible tangents.
Given: Circle O Construct: a tangent to circle O from P
3. Place compass point at midpoint of and stretch span to O or P.
4. Draw circle.
5. Connect P to where the two circles intersect to create tangents.
Why does this work?
Proof of Construction:
When the construction is finished, connect O to A to form a radius of circle O.
This radius also forms ΔOAP in circle M. Since is the diameter of circle M, ∠OAP is an angle inscribed in a semicircle, making it a right angle.
Since ∠OAP is a right angle, . In circle O, we now have the radius perpendicular to a line passing through a point on the circle (A), making a tangent to circle O.
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.
. In circle O, we now have the radius perpendicular to a line passing through a point on the circle (A), making 
a tangent to circle O.