How to find the length between 2 points given a pivot I am not great at math but I have done the previous steps to my problem. This is the last step where I need to find out the distance between C,D. 
I am writing a program that will output this distance so I need a formula that can calculate this based on the information given. If this is not possible are there any other positions or points that you would need to solve this?
Any help would be greatly appreciated.

------- EDIT -------
Firstly thank you for taking the time to answer, secondly i have to apologize for my diagram which was a little lacking in details. Here is a more clear version of what I intend to do with the formula. It is a door on a hing with a handle. The handle must clear the door jam when opened. I need to be able to calculate the distance of the clearance from C to D. If anything is unclear I will do my best to explain.
AE = 800
GB = 64.5
FB = 79.820508...(determined from another calculation that i did)
HD = 55 (40 door width + 15 offset for hing center)
CD = ??
Other than FB, all other dimensions on the image are known static numbers.

 A: let the radius of the circle be $R.$ we can compute $R$ in the following way:   $$R^2 = AB^2 = (AE-BF)^2 + (GB+ED)^2 = (800-79.82)^2 + (64.5+55)^2\\=532939.48 \to R = 730.02 $$ 
let $t$ be the angle the radius $AC$ makes with line parallel to $EF.$ then $$\cos t = \frac{ED}{R} = 0.07534, t = 1.49538$$
finally, $$CD = AE - R\sin t=800 - 730.02\sin t= 72.0547. $$
A: I will be making the following assumptions, which seem to be intended given the diagram:


*

*Point $A$ is at the center of the circle

*Point $B$ is on the circumference of the circle

*$AE \perp EF$, $DG \perp EF$, $FB \perp EF$, and $BG \perp BF$
Since $AE$ and $DG$ are both perpendicular to the same line, they are parallel. Draw a line parallel to $GB$ through $C$ intersecting $AE$ at $I$. Then triangle $IAE$ is a right triangle and
$$|IA| = \sqrt{r^2 - |ED|^2}$$
by the Pythagorean theorem. Now if we draw a line through $A$ parallel to $BG$ to intersect $DG$ at a point $A'$, $|A'C| = |IA|$, since we have a parallelogram (a rectangle, in fact) $AICA'$ and opposite sides of a parallelogram are equal. Now segment $AE$ equals segment $A'D$ as $AEDA'$ is also a parallelogram, so subtract $|A'C|$ from $|A'D|$ to get $|CD|$. Thus
$$|CD| = |A'D| - |A'C| = |AE| - |IA| = |AE| - \sqrt{r^2 - |ED|^2}$$.
To get the radius $r$ of the circle, Assumption 2 says that $B$ lies on it. The length of the segment from $A$ to the line $GB$ is given by subtracting $|EJ|$ from $|AE|$, where $J$ is the intersection point of lines $AE$ and $GB$. But $|EJ| = |FB|$, by another parallelogram argument. The length of the segment $JB$ is then $|JB| = |JG| + |GB|$ by yet another such argument. Now $|JG| = |ED|$ by a final parallelogram argument. Then by the Pythagorean theorem, the radius is the hypotenuse of the right triangle $\triangle JAB$, so $r^2 = (|AE| - |FB|)^2 + (|ED| + |GB|)^2$. Now just "Plug and Chug" to get the answer.
