Something like this?
\documentclass[tikz, border=5]{standalone}
\begin{document}
\begin{tikzpicture}
\path (120:3) coordinate (A) (0:3) coordinate (B) (0:0) coordinate (C);
\draw (A)
-- (B) node [at start, above left] {$A$} node [midway, above] {$c$}
-- (C) node [at start, right] {$B$} node [midway, below] {$a$}
-- (A) node [at start, below] {$C$} node [midway, below] {$b$}
-- cycle;
\draw [dashed] (A) |- (C) node [midway, below left] {$P$};
\draw (0:.5) arc (0:120:.5) (60:.4) -- (60:.6);
\end{tikzpicture}
\end{document}

or this...
\documentclass[tikz, border=5]{standalone}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\path (80:3) coordinate (A) (-30:3) coordinate (B) (0:0) coordinate (C);
\draw (A)
-- (B) node [at start, above left] {$A$} node [midway, right] {$c$}
-- (C) node [at start, right] {$B$} node [midway, below] {$a$}
-- (A) node [at start, below] {$C$} node [midway, left] {$b$}
-- cycle;
\coordinate (P) at ($(B)!(A)!(C)$);
\draw [dashed] (A) -- (P) node [below left] {$P$} -- (C);
\draw ($(C)!0.5cm!(B)$) arc (-30:80:.5cm) node [midway, sloped] {$|$};
\end{tikzpicture}
\end{document}

Your code produces errors; I tried to polish it by guessing what you were trying to do:

The code:
\documentclass[10pt]{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}[dot/.style={fill,circle,inner sep=1.5pt}]
\path
(80:5) node [dot,label=above left:$A$]{} coordinate (A)
(80:7) coordinate (a)
(20:9) node [dot,label=below:$B$]{} coordinate (B)
(20:11) coordinate (b)
(0:0) node[dot,label=below left:$P$]{} coordinate(P)
(-100:1)coordinate (e)
(-160:1) coordinate (f);
\node[dot,label=below:$Q$] (Q) at ($(P)!(A)!(B)$) {};
\node[dot,label=above right:$R$] (R) at ($(P)!(Q)!(A)$) {};
\draw[<->] (a) -- (e);
\draw[<->] (b) -- (f);
\draw[purple!70!black,dashed] (A) -- (Q);
\draw[green!70!black,dashed] (Q) -- (R);
\tkzMarkRightAngle(A,Q,P);
\draw[|<->|] ($(P)!3mm!90:(A)$)--node[fill=white,sloped] {$r$} ($(A)!3mm!-90:(P)$);
\draw[|<->|] ($(P)!-7mm!90:(Q)$)--node[fill=white,sloped] {$x$} ($(Q)!-7mm!-90:(P)$);
\draw[|<->|] ($(Q)!-3mm!90:(A)$)--node[fill=white] {$y$} ($(A)!-3mm!-90:(Q)$);
\tkzMarkAngle[size=0.75cm,color=cyan,mark=||](B,P,A);
\tkzMarkAngle[size=1cm,color=cyan,mark=|](P,A,Q);
\end{tikzpicture}
\end{document}
Best Answer
Without any fancy techniques:
Adopting the code from Torbjorn's answer: