Ruler and Compass Construction 27
Tangents to a Circle from an Outside Point

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1. Segment AB
2. Arcs centered at A, arbitrary radius
3. Arcs centered at B, same radius
4. Perpendicular bisector CD of AB
5. Circle centered at E, through A
6. Tangents BF, BG
7. Radius AF, showing right angle at F due to triangle inscribed in a semicircle
Why it works: The tangent line is perpendicular to the radius, and an angle inscribed in a semicircle is a right angle, so we just construct the circle with diameter AB, and its intersections with the original circle are the points of tangency.
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