### Given a point outside a circle, to construct a line
through the point tangent to the circle.

Given a circle and a point outside of the circle,

Connect the point, *A*, with the center of the circle, O

Let *M* be the midpoint of *OA*. Draw the circle centered at *M* going
through *A* and *O*.

Let the point where the two circles meet be *C*. Connect *AC*.

*AC* is a line through *A* tangent to the circle.

Top see that *AC* is tangent to the circle, connect *OC*

__/__*ACO* is an inscribed angle in the circle about *M*, so the
inscribed angle __/__*ACO* is half of the central angle which is the
diameter *AMO*. Since the inscribed angle is half as big as the central angle, it follows that __/__*ACO* is a right angle , and *OC* is perpendicular to *AC*. Since the tangent is
perpendicular to the radius to the point of tangency, by the
uniqueness of the line through *C* perpendicular to *OC*, *AC* is tangent
to the circle.

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