\documentclass[12pt]{amsart}
\usepackage{fullpage}
\usepackage{amsmath,tikz,fullpage,amsthm}
\usepackage{amssymb}
\usetikzlibrary{calc,intersections}
\begin{document}
\begin{tikzpicture}[bullet/.style={coordinate,after node path={(\tikzlastnode) node {$\scriptstyle \bullet$}}}, scale = 2]
 \draw [densely dotted,black] (0,0) circle (1cm);
 \coordinate[label=below left:$A$] (A) at (-.87,-.5);
 \coordinate[label=below right:$B$] (B) at (.87,-.5);
 \coordinate[label=above:$C$] (C) at (-.45,.9);
 \draw [semithick,black] (A) -- (B) -- (C) -- cycle;
 \path[name path=alta] (A) -- ($(B)!(A)!(C)$);
 \path[name path=altb] (B) -- ($(A)!(B)!(C)$);
 \draw[name intersections={of=alta and altb,by=x}] (A) -- (x) node[fill=white,pos=0.6,inner xsep=0pt,inner ysep=1.5pt,right=-7pt] {\small $\cos \alpha$};
 \draw (B) -- (x) node[fill=white,pos=0.5,inner xsep=0.5pt] {\small $\cos \beta$};
 \draw (C) -- (x) node[fill=white,pos=0.7,inner xsep=0pt,right=-7pt] {\small $\cos \gamma$};
\end{tikzpicture}
\end{document}