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.
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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 $AB$.