Theorem 5.1: Two consecutive points in a partition of an arc can not be diametrically opposed.
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: We can't have two adjacent points in a partition diametrically opposed, because if we did,
then since P1 and P2 are on the same line, on opposite sides of O, P2 is on the opposite side of OP3 than P1 by Theorem 2.5, so it is not inside the angle between OP1 and OP3.
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 the line determined by OA by Theorem 3.7. Similarly, the line which is tangent to the circle at B is perpendicular to the line determined by 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.