Theorem: (Tangent - Chord Equivalence)

If two lines are drawn from extension point P to circle, C, so that one is tangent at point T and the other passes through points A and B on the circle then PT x PT = PA x PB.

Proof:

Let there be a circle, C, with an exterior point P.

Also, let there be a line drawn from point P so that it is tangent to the circle at point T.

Also draw a line from point P that passes through points A and B which lie on the circle.

Draw two chords, one from point T to point A and the other from point T to point B.

Let Q be a point on the tangent line so that T is between the points P and Q.



By the Tangent - Chord and Inscribed Angle Theorem, the measure of angle PBT equals the measure of angle PTA.

Similarily, the measure of angle TAB equals the measure of angle BTQ.

The angles PAT + TAB = 180 and angles BTQ + BTP =180.

Since TAB equals BTQ, PAT equals BTP.

By corresponding angles triangles PTB and PAT are similar.

Since the triangles are similar, we can use the proportion: PA / PT = PT / PB

Hence, PT² = PA x PB.