Construction #7

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.