Need help with finishing a picture using tikz

Question

\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\geometry{verbose}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}
\begin{document}
\begin{center}
\begin{tikzpicture}[scale=0.9]
\draw[thick] (0,0) -- (0,3) -- (4,0)-- (0,0); 
\filldraw[thick, fill=white] (0,3) arc (90:270: 1.5);
\filldraw[thick, fill=white] (0,0) arc (180:360: 2);
\filldraw[black] (0,0) node[below left] {$B$};
\filldraw[black] (0,3) node[above left] {$A$};
\filldraw[black] (4,0) node[anchor=west] {$C$};
\end{tikzpicture} 
\end{center}
\end{document}

I have this so far to make the triangle and label the points but how do you draw the semicircles??

Answer 1

I will be happy to delete this but since I had this slight extension of this answer on my machine I could also just post it. This defines a style for the semicircle that measures the length and angle of the path and puts the arc accordingly. If you want it to be on the other side of the path, add yscale=-1.

\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{center}
\begin{tikzpicture}[scale=0.9,pics/semicircle/.style={code={
 \draw (0,0) arc[start angle=180,end angle=0,radius=#1/2];}},
 semicircled/.style={to path={let \p1=(\tikztostart),
  \p2=(\tikztotarget),\n1={veclen(\x1-\x2,\y1-\y2)},
  \n2={atan2(\y2-\y1,\x2-\x1)} in 
  (\p1) pic[rotate=\n2,transform shape]{semicircle=\n1}
  }},bullet/.style={fill,inner sep=1.2pt,circle}]
  %
  \draw (0,3) node[bullet,label=above left:{$A$}](A){}
    -- (0,0) node[bullet,label=below left:{$B$}](B){}
    edge[semicircled] (A)
    --  (4,0) node[bullet,label=below right:{$C$}](C) {}
    edge[semicircled] (B)
    edge[semicircled,yscale=-1] (A)
  --cycle;
\end{tikzpicture} 
\end{center}
\end{document}