How to match the blue-violet line with the orange line in this tikz picture

Question

I have this diagram,

And the part I am only concerned with is the lines joining the centers C2 (blue circle) and C3 (violet circle). No matter how much values I change in the following code, they don't coincide. I am not sure why. Note that the lines joining the points of tangency (from l line) and the centers as well as point of contact between the circles were obtained by trial and error with the help of intersections as you will see in the code. Also please suggest me if there is any other better way to get the same diagram.

Here's the code:

%% Figure 13
\documentclass[border=3mm,tikz]{standalone}
\usepackage{tkz-euclide}
\usetikzlibrary{intersections,calc}
\begin{document}
\begin{tikzpicture}[scale=0.9]
%frames
\draw[thick,->] (0,0) coordinate (origin) -- (23,0) coordinate (a1) node[right]{$x$}; % x-axis
\draw[thick,->] (origin) -- (0,20) coordinate (a2) node[above]{$y$}; % y-axis
\draw[name path=lline,thick] (origin) -- (60:22.25) coordinate (a3) node[anchor=south west]{$l$}; % l-line
%circle 1
\coordinate (CC1) at (1.732,1){};
\coordinate[label={[red]below:$C_1$}] (C1) at (1.732,0){};
\tkzDrawPointred,scale=2pt
\tkzDrawCirclename path=circle1,red;
\drawthick,red--(CC1) node[right,pos=.5]{1};
\draw[name intersections={of=lline and circle1}] (intersection-1) coordinate (A1);
\drawthick,red--(CC1) node[above,pos=.5]{1};
%circle 2
\coordinate (CC2) at (5.1983,3){};
\coordinate[label={[blue]below:$C_2$}] (C2) at (5.1983,0){};
\tkzDrawPointblue,scale=2pt
\tkzDrawCirclename path=circle2,blue;
\drawthick,blue--(CC2) node[right,pos=.5]{3};
\draw[name intersections={of=lline and circle2}] (intersection-1) coordinate (A2);
\drawthick,blue--(CC2) node[above,pos=.5]{3};
%circle 3
\coordinate (CC3) at (15.59229,9){};
\coordinate[label={[violet]below:$C_3$}] (C3) at (15.59229,0){};
\tkzDrawPointviolet,scale=2pt
\tkzDrawCirclename path=circle3,violet;
\drawthick,violet--(CC3) node[right,pos=.5]{$r$};
\draw[name intersections={of=lline and circle3}] (intersection-1) coordinate (A4);
\drawthick,violet--(CC3) node[above,pos=.5]{$r$};
% circle 1 and circle 2 point of contact
\draw[name intersections={of=circle1 and circle2}] (intersection-1) coordinate (A3);
% circle 2 and circle 3 point of contact
\draw[name intersections={of=circle2 and circle3}] (intersection-1) coordinate (A5);
%bisector
%\tkzDrawBisector(a1,origin,a3)
\draw[red,thick] (origin) -- (CC1);
\drawred,thick -- (A3);
\drawblue,thick -- (CC2);
\drawblue,thick -- (A5);
\drawviolet,thick -- (CC3);
%angle marks 1
\tkzMarkAnglesize=.6,mark=none
\tkzLabelAnglefill=white,dist=.2{$2\alpha$}
%angle marks 2
\tkzMarkAnglesize=1,mark=none,red
\tkzLabelAngledist=.8,red{$\alpha$}
\tkzMarkRightAnglesize=.2,red
%angle marks 3
\tkzMarkRightAnglesize=.2,blue
%angle marks 4
\tkzMarkRightAnglesize=.2,violet
%markings
\draw[orange,very thick] (CC2) --++ (0:10.4) coordinate (a5);
\draw[orange,very thick] (CC3) -- (a5);
\draw[orange,very thick] (CC2) -- (CC3);
\end{tikzpicture}
\end{document}

UPDATE For some reasons, no matter how accurate values you give in the coordinates, there will always be some discrete points, meaning that there isn't actually 'continuity' between each points, so after a certain increment of a value of coordinates, it will switch to another discrete point. It may be due to the way the computer or the system works.

Answer 1

I'd to this with a pic for the circles with the radii, the right angles and the labels. Also if the origin of the pic is at the origin of coordinates it will be easier to draw.

Something like this:

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{calc}
\tikzset
{%
    pics/my circle/.style n args={4}{% #1 = radius, #2 = tangent angle,
    code={%                            #3 = radius label, #4 = circle label
      \pgfmathsetmacro\aux{90+2*#2}          % angle, top radius
      \coordinate (-C) at ({#1/tan(#2)},#1); % center
      \draw (-C) circle (#1);
      \fill (-C) circle (3pt);
      \node at ({#1/tan(#2)},0) [below] {$#4$};
      \foreach\i in {-90,\aux}
      {
        \begin{scope}[shift={(-C)},rotate=\i]
          \draw (0,0) -- (#1,0) node[midway,yshift=1.5mm,xshift=1mm] {$#3$};
          \draw ($(#1,0)+(0,-0.3)$) --++ (-0.3,0) --++ (0,0.3);
        \end{scope}
      }

    }},
}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round]
% circles
\pic[red]     (C1) {my circle={1}{30}{1}{C_1}};
\pic[blue]    (C2) {my circle={3}{30}{3}{C_2}};
\pic[magenta] (C3) {my circle={9}{30}{r}{C_3}};
% axes
\draw[latex-latex] (0,18) node [above] {$y$} |- (25,0) node [right] {$x$};
% tangent, bisector
\draw (0,0) --++ (60:20) node [above right] {$\ell$};
\draw (0,0) --++ (30:20);
% triangle
\draw[very thick,orange] (C2-C) -- (C3-C) |- cycle;
% angles
\draw      (0.3,0) node[below] {\strut$2\alpha$} arc (0:60:0.3);
\draw[red] (0.7,0) node[below] {\strut$\alpha$} arc (0:30:0.7);
\end{tikzpicture}
\end{document}