Zentrierungsproblem und bessere Nodesbeschriftung

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\begin{figure}[H] %Skizze zu Beweis mit Supportlines
\centering
\begin{tikzpicture}[auto,scale=1.17]
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=right:$B$] (B) at (12,0);
\coordinate [label=above:$C$] (C) at (60:12);
\coordinate [label=above:$0$] (P) at (6,3);
\coordinate (D) at ($(A)!(P)!(B)$); %D is perpendicular from 0 to AB
\coordinate (E) at ($(B)!(P)!(C)$); %E is perpendicular from 0 to BC
\coordinate (F) at ($(C)!(P)!(A)$); %F is perpendicular from 0 to CA
\node (X) at ($ (A)!.56!(B) $) {}; %X auf 0.6 von AB
\node (Y) at ($ (B)!.58!(C) $) {}; %Y auf 0.6 von BC
\node (Z) at ($ (C)!.62!(A) $) {}; %Z auf 0.6 von CA
%draws triangle ABC and labels l-overline
%\draw (A) -- node[below]{$\overline{l}(\alpha+2\pi/3)$} (B)  -- node[right]{$\overline{l}(\alpha+4\pi/3)$} (C) -- node[left]{$\overline{l}(\alpha)$} (A);
\draw (A) -- (B) -- (C) -- cycle; 
\node (a) at ($ (B)!.35!(C) $) {}; %a für Beschriftung von l-overline
\node (b) at ($ (C)!.4!(A) $) {}; %b für Beschriftung von l-overline
\node (c) at ($ (A)!.35!(B) $) {}; %c für Beschriftung von l-overline
\node at (a) [right] {$\overline{l}(\alpha+4\pi/3)$};
\node at (b) [left] {$\overline{l}(\alpha)$};
\node at (c) [below] {$\overline{l}(\alpha+2\pi/3)$};
%draws dashed q-lines from 0 to inscribing triangle
\draw[style=densely dashed] (D) -- node[left] {$q(\alpha+\frac{2\pi}{3})$} (P); 
\draw[style=densely dashed] (E) -- node[below] {$q(\alpha+\frac{4\pi}{3})$} (P);
\draw[style=densely dashed] (F) -- node[above] {$q(\alpha)$} (P);
%draws convex closed curve
\draw[thick]
    (X) to[out=0,in=-60] (Y) to[out=120,in=60] (Z) to[out=-120,in=180] (X); %leaves X with angle 0, goes in to Y with angle -60 and so on
%draws l lines in blue
%l line perpendicular to AB
\coordinate (G) at ($ (X)!10cm!90:(B) $);
\coordinate (H) at ($ (X)!1.3cm!270:(B) $);
\draw [blue] (G) -- (H);
\node at (H)[blue, below] {$l(\alpha+\frac{2\pi}{3})$}; 
%l line perpendicular to BC
\coordinate (I) at ($ (Y)!9.5cm!90:(C) $);
 
 
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