AlterMundus

Means

Example 1

\documentclass{standalone}
\usepackage{tkz-euclide,mathtools} 
\usetikzlibrary{math}
\tkzSetUpColors[background=white,text=black]  
\tkzSetUpPoint[size=2,color=teal]
\tkzSetUpLine[very thin,color=teal]
\tkzSetUpCompass[color=orange,ultra thin,/tkzcompass/delta=10] 

\parindent=0pt
\begin{document}
\begin{tikzpicture}[scale=4]
  \tikzmath{
  \a = 3 ;
  \b = 1; }
  \tkzDefPoint(0,0){O}
  \tkzDefPoint(-\a,0){A}
  \tkzDefPoint(\b,0){B}
  \tkzDefMidPoint(A,B)                     \tkzGetPoint{I}
  \tkzDrawSemiCircle(I,B)
  \tkzDefPointWith[orthogonal normed](O,B) \tkzGetPoint{o}
  \tkzDefPointWith[orthogonal normed](I,B) \tkzGetPoint{i}
  \tkzInterLC(O,o)(I,B)                    \tkzGetFirstPoint{Mg}
  \tkzDefTangent[at=Mg](I)                 \tkzGetPoint{t}
  \tkzDuplicateSegment(I,O)(Mg,t)          \tkzGetPoint{Mq}
  \tkzDefPointBy[projection=onto Mg--t](O) \tkzGetPoint{Mh}
  \tkzMarkRightAngles[fill=teal!10, opacity=.40,size=.1](o,O,I I,Mg,Mh O,Mh,Mg)
  \tkzDrawSegments(A,B O,Mg I,Mq O,Mh I,Mg)
  \tkzDrawLine[add=1 and 0.25](Mg,t)
  \tkzDrawPoints(A,B,I,O,Mg) 
  \tkzDrawSegment[draw=none,dim={$a$,-24pt,below=6pt}](A,O)
  \tkzDrawSegment[draw=none,dim={$b$,-24pt,below=6pt}](O,B)
  \tkzDrawSegment[draw=none,dim={$\dfrac{a-b}{2}$,-48pt,below=6pt}](I,O)
  \tkzDrawSegment[draw=none,dim={$\dfrac{a+b}{2}$,-80pt,below=6pt}](I,B)
  \tkzLabelSegment[right](O,Mg){$\sqrt{ab}$}
  \tkzLabelSegment[right](I,Mq){$\sqrt{\dfrac{a^2+b^2}{2}}$}
  \tkzLabelSegment[right](O,Mh){$\dfrac{2ab}{a+b}$}
  \tkzMarkAngles[size=.2,mark=|](Mh,O,Mg I,Mg,O)
  \tkzFillAngles[size=.2,fill=red!20,opacity=.4](Mh,O,Mg I,Mg,O)
  \tkzMarkSegments[mark=||](Mg,Mq I,O)
\end{tikzpicture}
\end{document}

Example 2

Means at rozenblum.com

[PDF] [TEX]

\documentclass{standalone}
\usepackage{tkz-euclide,mathtools} 
\usetikzlibrary{math}
\tkzSetUpColors[background=white,text=black]  
\tkzSetUpPoint[size=2,color=teal]
\tkzSetUpLine[very thin,color=teal]
\tkzSetUpCompass[color=orange,ultra thin,/tkzcompass/delta=10]
\tikzset{new/.style={color=orange,ultra thin}}  

\parindent=0pt
\begin{document}
\begin{tikzpicture}[scale=1]
   \tikzmath{\a = 12 ;
   \b = 3 ;} 
   \tkzDefPoint(0,0){O}
   \tkzDefPoint(\a,0){A}
   \tkzDefPoint(\a-\b,0){B}
   \tkzDefMidPoint(O,B)                           \tkzGetPoint{I}
   \tkzDrawCircle(I,B)
   \tkzDefTangent[from = A](I,B)                 \tkzGetFirstPoint{Mg}
   \tkzDefPointWith[orthogonal normed](I,A)      \tkzGetPoint{i}
   \tkzInterLC(I,i)(I,B)                         \tkzGetFirstPoint{Mq}
   \tkzDefPointBy[projection=onto O--A](Mg)      \tkzGetPoint{Mh}
   \tkzMarkRightAngles[fill=teal!10,
        opacity=.40,size=.4](A,Mg,I A,I,Mq)
   \tkzDrawSegments(O,A A,Mg I,Mg Mg,Mh A,Mq I,Mq)
   \tkzDrawPoints(A,B,I,O,Mg,Mq)
   \tkzLabelPoints(A,B,I,O,Mg,Mq)
   \tkzLabelSegment[left=6pt](I,Mq){$\dfrac{a-b}{2}$}
   \tkzDrawSegment[draw=none,black,dim={\normalsize  
         $\dfrac{a+b}{2}$,-16pt,below=6pt}](I,A)
   \tkzLabelSegment[right=6pt](A,Mg){$\sqrt{ab}$}
   \tkzLabelSegment[below right](A,Mq){$\sqrt{\dfrac{a^2+b^2}{2}}$}
   \tkzMarkSegments[mark=s||](I,Mq I,O I,Mg)
\end{tikzpicture}
\end{document}