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

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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