Theorem 5.1: If two distinct points on a circle are not diametrically opposed, then the tangent lines at those points meet a point outside the circle.

Proof: Let   A   and   B   be the two points on the circle which are not diametrically opposed, and let   O   be the center of the circle.

If   OA   and   OB   determine the same line, since, by Theorem 3.3, a line and a circle meet at at most two points,   A   and   B   would either have to be the same point or be diametrically opposed, which we are hypothesizing does not happen. So, if   OA   and   OB   determine distinct lines which share a point, Then by Theorem 1.6, they cannot be parallel, so they have different slopes. But the line which is tangent to the circle at   A   is perpendicular to   OA   by Theorem 3.7. Similarly, the line which is tangent to the circle at   B   is perpendicular to   OB.   Since, by definition, perpendicular lines have slopes which are negative reciprocals of each other, the tangent lines have different slopes, and so are not parallel. Then by Theorem 1.6 they meet.

Let's call the point where they meet   C.   Since by Theorem 3.6, the shortest distance from   O   to   AC   is the perpendicular distance which is the distance from   O   to   A,   C is farther away, and by definition will lie outside the circle.

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Next Theorem (5.2)

Analytic Foundations of Geometry

R. S. Wilson