Constructions

Compass, Ruler

The construction of tangents from an external point requires joining points with straight lines (using a ruler) and drawing circles/arcs of specific radii, such as the circle on OP as diameter and marking equal radii (using a compass). A protractor or set square is not needed since no angles are measured directly and no perpendicular lines from a set square are used in this method. Hence only the compass and ruler are the correct tools.

Given circle with center O, Auxiliary circle with center M

Two circles are involved: the original given circle with center O, and the auxiliary circle with center at midpoint M of PO. These two circles intersect at points which help locate the tangent points.

1

From a point on the circle, exactly one tangent can be drawn. This tangent is perpendicular to the radius at that point.

2

The auxiliary circle intersects the given circle at exactly 2 points. These intersection points are the points of tangency for the two tangents from the external point.