I have a problem to understand the PPL Apollonius problem part as stated here on the page 202. My problem is, where is the diameter and the center of the Thalet circle as given here in the second case as $c$ with $a^2=c\cdot c_a$.Here, I believe $t=a$ and $c=|AP_1|$ and $c_a=|AP_2|$. I can construct the $t$ in question separately on a blank paper but I cannot insert it into the picture in the bottom of the page 202; I think that the Thalet cicrcle and its center is not designated in the picture from the snippet as below:
If I understand the question, the construction is not difficult. In Elements, VI, 13 Euclid constructs the (geometric) mean proportional between two given straight lines: place the lines end to end, make them the diameter of a circle, and erect a perpendicular at their meeting point; the segment between the diameter and the circumference is the mean proportional.
Thus, in the $PPL$ problem, if $B$, $C$ in the figure above are two points, and the line through them intersects the given line at $A$, and you want the mean proportional between external segment $AB$ and secant $AC$, make $AD=AB$ and construct a circle with diameter $DC$. The perpendicular $AE$ is the mean proportional, which can be laid off on the given straight line as $AF$, $AG$.
The intersections $H$, $J$, then, of perpendiculars at $F$, $G$ with the perpendicular bisector of $BC$ are the centers of two circles tangent at $F$, $G$ and passing through $B$, $C$.