Example 1 Proposition 1 of the Book of Lemmas

If two circles touch at $A$, and if $[CD]$, $[EF]$ be parallel diameters in them, $A$, $C$ and $E$ are aligned.


\tkzSetUpCompass[color=orange,ultra thin,/tkzcompass/delta=10]
\tikzset{label style/.append style={color=teal}}
\tikzset{new/.style={color=orange,ultra thin}} 

  \tkzInterLC(F,O_1)(O_1,A) \tkzGetSecondPoint{E}
  \tkzDefLine[parallel=through O_2](E,F) 
  \tkzInterLC(x,O_2)(O_2,A) \tkzGetPoints{D}{C} 
  \tkzDrawCircles(O_1,A O_2,A)
  \tkzDrawSegments[new](O_1,A E,F C,D)
  \tkzDrawSegments[purple](A,E A,F)

Example 2 Proposition 6 of the Book of Lemmas

Let $[AB]$, the diameter of a semicircle, be divided at $C$ so that $AC = \varphi·CB$ [or in any ratio]. Describe semicircles within the first semicircle and on $[AC]$, $[CB]$ as diameters, and suppose a circle drawn touching the all three semicircles. If $[GH]$ be the diameter of this circle, to find relation between $GH$ and $AB$.