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.

[PDF] [TEX]

\documentclass{standalone}
\usepackage{tkz-euclide}
\tkzSetUpPoint[size=1,color=teal]
\tkzSetUpLine[thin,color=teal]
\tkzSetUpCompass[color=orange,ultra thin,/tkzcompass/delta=10]
\tikzset{label style/.append style={color=teal}}
\tikzset{new/.style={color=orange,ultra thin}}

\begin{document}
\begin{tikzpicture}
\tkzDefPoints{0/0/O_1,0/1/O_2,0/3/A}
\tkzDefPoint(15:3){F}
\tkzInterLC(F,O_1)(O_1,A) \tkzGetSecondPoint{E}
\tkzDefLine[parallel=through O_2](E,F)
\tkzGetPoint{x}
\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)
\tkzDrawPoints(A,O_1,O_2,E,F,C,D)
\tkzLabelPoints(A,O_1,O_2,E,F,C,D)
\end{tikzpicture}
\end{document}

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 $AB$.

[PDF] [TEX]