### Steve Wilson

We are given the circle with center at O, and a point A outside the circle. Connect AO, and let M be its midpoint.

Draw the circle centered at M going through O. Since M is the midpoint of AO, it will also go through A. Let B and C be the points where this circle intersects the given circle. Then AB and AC will be the lines through A tangent to the given circcle.

This is because /ABO and /ACO will be inscribed angles in the circle centered at M, so their measure will be half of the central angle. But since AMO is a diameter of the circle, the central angle is a straight angle, so /ABO and /ACO will be right angles. Since the tangent is perpendicular to the radius at the point of tangency, AB and AC will be tangents to the circle.

This construction will help us use the Isosceles Triangle Theorems to prove a theorem about tangents.

Theorem: Given a circle and a point outside the circle, the line from the point to the center of the circle will bisect the angle formed by the tangents, and the point is equidistant from the two points of tangency.

Proof: BC will be perpendicular to AO because the line that joins the two points where two circles intersect will be perpendicular to the line that joins their centers. MB and MC will be congruent, because they are both radii of the same circle. Therefore triangle MBC is isosceles, and the line that is perpendicular to the base will bisect the vertex angle. Thus, /BAO and /CAO which are inscribed angles for these congruent angles will also be congruent, and AO will bisect the angle formed by the tangents.

In triangle ABC the line which bisects the vertex angle is perpendicular to the base, so the triangle is isosceles.

It is possible to prove these results simply using congruent triangles, but this construction enables us to use the Isosceles Triangle Theorems.