Lemma 6: C is on the line segment between E and F.
Proof:
By Lemma 3, E is on the line segment between H and A, so by Theorem 2.7, E is on the same side of OH as A.
In Lemma 1 we proved that OC intersected AB at a point G on the line segment between A and B, so by Theorem 2.4, A is on one side of G and B is on the other, so by Theorem 2.5, A is on the other side of OH from B.
If D is not on the line determined by O and H, then since H is on the line segment between A and D, by Theorem 2.4, D would be on the other side of H from A, and then by Theorem 2.5, D would be on the other side of OH from A, so D is on OH or on the same side of OH as B.
By Theorem 2.7, every point between D and B including, by Lemma 5, F is on the same side of OH as B, so F is on the other side of OH from E.
Conclude by Theorem 2.8 that C in on the line segment between E and F.