How does this derivation of a geometric/quadratic formula for a conic, have a multiple of 4 in it?

Question

Here is a diagram from an old book that I'm going through:

What I'm unable to understand how is $XG^2 = 4(AB.BG)$

Here's what's given about the construction:

$AX = AQ = MG$ and $BG = BQ$

$\square(ABEF)$ is a square and so is $\square(ARSK)$ where $AK = 2.AF$

$G$ is any point between $A$ and $B$ and $MG$ is perpendicular to $AB$

Given this, how does one get:

$XG^2 = 4(AB.BG)$?

I've tried it quite a few ways and substituted many things, but seem to be missing something obvious perhaps.

Here's how the manuscript lays it out:

It somehow just concludes the relation without really clarifying where it came from. It does look like a mean proportional (i.e., geometric mean) in disguise, but can't put a finger on its derivation.

PS: Not a homework problem. Purely recreational curiosity.

Answer 1

Here's a one-circle solution. (See the edit history for an alternative.)

$$|\overline{GX}|^2 = |\overline{QG}||\overline{Q^\prime G}| = 2a \cdot 2(a+b)= 4 \; |\overline{BG}||\overline{AB}|$$