I need to make a circle that is exactly like the one on the first page of http://www.math.siu.edu/kocik/papers/metaptolemiana.pdf in Figure 1 (the right hand circle) here is all I could conjure up:

\begin{tikzpicture}[bullet/.style={coordinate,after node path=
{(\tikzlastnode) node {$\scriptstyle \bullet$}}}, scale = 2]
\draw [densely dotted,black] (0,0) circle (1cm);
\draw [semithick,black] (-.87,-.5) -- (-.45,.9) -- (.87,-.5) -- cycle;
\coordinate[label=above:$A$] (A) at (-1.02,-.66);
\coordinate[label=above:$C$] (C) at (-.45,.92);
\coordinate[label=above:$B$] (B) at (1.00,-.66);

I need help drawing the three altitudes (lines from vertices) and being able to add cos(x) like it does in the picture. I need help the most on what I underlined. I am not very knowledgeable in tikz. I just know the very very basics. The labeling of the vertices I googled. Or does anyone know of any program that I could make that circle appear that way and then it would convert it to Tikz/Latex ?