If two circles touch at $A$, and if $[CD]$, $[EF]$ be parallel diameters in them, $A$, $C$ and $E$ are aligned.
View TeX code
\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}
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 $AB$.
View TeX code
\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}}
\tikzset{step 1/.style={color=cyan,ultra thin}}
\tikzset{step 2/.style={color=purple,ultra thin}}
\begin{document}
\begin{tikzpicture}[scale=1, /pgf/fpu/install only={reciprocal}]
\tkzDefPoints{0/0/A,10/0/B}
\tkzDefGoldenRatio(A,B) \tkzGetPoint{C}
\tkzDefMidPoint(A,B) \tkzGetPoint{O_1}
\tkzDefMidPoint(A,C) \tkzGetPoint{O_2}
\tkzDefMidPoint(C,B) \tkzGetPoint{O_3}
\tkzDefExtSimilitudeCenter(O_2,A)(O_3,C) \tkzGetPoint{M_0}
\tkzDefIntSimilitudeCenter(O_1,A)(O_2,A) \tkzGetPoint{M_1}
\tkzDefIntSimilitudeCenter(O_1,B)(O_3,B) \tkzGetPoint{M_2}
\tkzInterCC(O_2,A)(M_2,B) \tkzGetFirstPoint{P_2}
\tkzInterCC(O_3,B)(M_1,A) \tkzGetSecondPoint{P_3}
\tkzInterCC(O_1,A)(M_0,C) \tkzGetFirstPoint{P_1}
\tkzInterLL(O_2,P_2)(O_3,P_3) \tkzGetPoint{O_4}
\tkzInterLC[common = P_1](A,P_1)(O_4,P_1) \tkzGetFirstPoint{G}
\tkzInterLC[common = P_1](B,P_1)(O_4,P_1) \tkzGetFirstPoint{H}
\tkzDefPointsBy[projection = onto A--B](O_4,G,H){P,F,E}
\tkzInterLL(B,G)(H,E) \tkzGetPoint{M}
\tkzInterLL(A,H)(G,F) \tkzGetPoint{L}
\tkzInterLC[common = C](M,C)(O_3,C) \tkzGetFirstPoint{K}
\tkzInterLC[common = C](L,C)(O_2,C) \tkzGetFirstPoint{I}
\tkzDrawCircles(O_4,P_2)
\tkzDrawSemiCircles[](O_1,B O_2,C O_3,B)
\tkzDrawSegments(A,B)
\tkzDrawSegments[step 1](O_4,P O_4,O_1 O_4,O_2 O_4,O_3)
\tkzDrawSegments[new](A,P_1 P_1,B B,G A,H O_1,P_1)
\tkzDrawSegments[new](C,H C,I C,K)
\tkzDrawSegments[step 2](F,G G,H H,E O_1,P_1)
\tkzDrawPoints(A,B,C,O_4,P_1,G,H,P,O_1,O_2,O_3,F,E,L,M,P_3,P_2,K,I)
\tkzLabelPoints(A,B,C,G,H,P,O_4,O_1,O_3,L,M,P_3,P_2,K,I)
\tkzLabelPoints[above](P_1)
\tkzLabelPoints[below](O_2)
\tkzLabelPoints[above right](E)
\tkzLabelPoints[above left](F)
\tkzMarkRightAngles[fill=teal!15,opacity=.4](B,F,G O_1,P,O_4 H,E,P)
\end{tikzpicture}
\end{document}