Theorem 5.2

Theorem 5.2: Let P be a partition of the arc between A and B . Then the inside approximation corresponding to P is shorter than the outside approximation.

If Q is a refinement of P, then the inside approximation corresponding to Q is larger than the inside approximation corresponding to P, and the outside approximation corresponding to Q is less than the outside approximation corresponding to Q.

Proof: By Theorem 5.1, the tangents will meet at points T₁, T₂, . . . T_n, so we can take the outside approximation.

The inside approximation is shorter than the outside approximation because since a straight line is the shortest distance between two points, by Theorem 3.5, the distance going straight from P_i to P_i+1 will be shorter than the distance going through T_i+1, provided that we can show that P_i is on the segment between T_i and T_i+1 so that we could use Theorem 2.3. For this we will need to first get some information concering the intersections of the tangents. To do this it will suffice to consider a partition which has three points where the first and last points are not diametrically opposed.

For the assertion concerning the refinement, it suffices to show that the result will hold if we add one extra point to the partition.

The figure above can be used to show that C is on the line segment between A and B, that the inside approximation going from A to C to B will be longer than the inside approximation going straight from A to B, and that outside approximation corresponding to A, B, C is shorter then the outside approximation using just A and B.

The inside approximation going from A to C to B will be longer than the inside approximation going straight from A to B because, by Theorem 3.5, the shortest distance between two points is a straight line.

Similarly, the distance from E straight across to F will be shorter than the distance going from E through D to F. So we need to prove that E is on the line segment between A and D, and F is on the line segment between D and B, and then we could apply Theorem 2.3 to conclude that the outside approximation corresponding to A, B, C is shorter then the outside approximation using just A and B.

Given:

C is on the arc of the circle in the angle between OA and OB.
D is the point where the tangent to the circle at A meets the tangent to the circle at B.
E is the point where the tangent to the circle at A meets the tangent to the circle at C.
F is the point where the tangent to the circle at B meets the tangent to the circle at C.

To prove:

We will prove the following lemmas

Lemma 1: The line determined by OC intersects the line determined by AB at a point G which is on the line segment between A and B.
Lemma 2: There is a point H on the line determined by OC which is either on the line segment between A and D, on the line segment between D and B, or D is on the line determined by OC.
Lemma 3: E is on the line segment between A and H.
Lemma 4: E is on the line segment between A and D.
Lemma 5: F is on the line segment between D and B.
Lemma 6: C is on the line segment between E and F.