What are the formal terms for the intersection points of the geometric representation of the extended trigonometric functions? Mike Pierce's answer to this question, regarding trigonometric functions beyond the common (co)sine, (co)secant, and (co)tangent, points to a figure on the Wikipedia page on trigonometric functions that nicely illustrates the relationships between the magnitudes of some of these various additional functions [(co)versine, ex(co)secant], for a given angle $\theta$:

The point $O$ in the figure is the center of the circle. What, if any, are the formal names for the other labeled points, $A$ through $F$?  The names for the various line segments and areas are relatively easy to find on Wikipedia (radii, chords, tangent, etc.), but I haven't been able to track down any names for these points.
A book cited in a comment in the above-linked question, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, may contain the answers, but I don't have a copy on hand.
 A: This might not exactly answer your question, but $E$ is the inversion of $C$ in the circle centered at $O$ with radius $OA$, while the point $E$ is the pole of the extended line $AB$ in the same circle (projective geometry).
Because $E$ is the inversion of $C$, we can say that $OE \cdot OC = R^2 = 1$ (the converse also holds). Likewise, as $\sin{\theta}\cdot \csc{\theta} = 1$, those two points are inverses as well (the points $F$ and an unnamed point in the figure). Note that inverse points are defined as being collinear with the center of the circle.
This link is about projective geometry, and this one is about inversion. Inversion is based on elementary geometry (similarity of triangles, homothecy, etc.) so it is comparatively easier to use and understand as compared to projective geometry, which uses a lot of different theory.
A: The given circle is the unit circle, therefore $O$ can also be called the origin. I don't think any of $A,B,C,E$ have a name, since they are depedent on $\theta$, so it can be any point on the circle (in the case of A and B) or on the x-axis (in case of C and E). I suspect that $D$ actually has a name, because it is an important point on the unit circle, but I couldn't find it. 
