圓與線的交點「路徑」（二）

Question

按照這個答案，對於用 (1) 三點定義圓的情況，是否也可以使用類似的樣式來找出交點； (2) 一個點和一個中心？

\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
    \coordinate (A) at (0,0);
    \coordinate (B) at (2,5);
    \coordinate (C) at (4,0);
    \coordinate (D) at (6,3);
    \coordinate (E) at (4,2);

    \draw (A)--(B)--(C)--cycle (D)--(E);
\end{tikzpicture}
\end{document}

（例如，（1）經由A、B、C的圓與線段DE的交點；（2）經由A與圓心B的圓與線段DE的交點。）

Answer 1

對於情況（1）：基於此很好的答案,

對於情況（2）：使用簡單的through函式庫。

\documentclass[margin=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections,through,calc}
\tikzset{circle through 3 points/.style n args={3}{%
insert path={let    \p1=($(#1)!0.5!(#2)$),
                    \p2=($(#1)!0.5!(#3)$),
                    \p3=($(#1)!0.5!(#2)!1!-90:(#2)$),
                    \p4=($(#1)!0.5!(#3)!1!90:(#3)$),
                    \p5=(intersection of \p1--\p3 and \p2--\p4)
                    in },
at={(\p5)},
circle through= {(#1)}
}}
\begin{document}

\begin{tikzpicture}
    \coordinate (A) at (0,0);
    \coordinate (B) at (2,5);
    \coordinate (C) at (4,0);
    \coordinate (D) at (6,3);
    \coordinate (E) at (4,2);
    \draw (A)--(C)node[midway,below]{(1)}--(B)--cycle;
    \path [draw, name path=line] (D)--(E);
    \node[name path=circ, circle through 3 points={A}{B}{C},draw=blue]{};
    \path [name intersections={of=circ and line, by={K}}] ;
    \node[circle,minimum size=2pt,fill=red] at(K) {};

\end{tikzpicture}
\hfill
\begin{tikzpicture}[scale=0.55]
    \coordinate (A) at (0,0);
    \coordinate (B) at (2,5);
    \coordinate (C) at (4,0);
    \coordinate (D) at (6,3);
    \coordinate (E) at (4,2);
    \draw (A)--(B)--(C)--cycle;
    \path [draw, name path=line] (D)--(E);
    \node [draw, name path=circ] at (A) [circle through=(B)] {};
    \path [name intersections={of=circ and line, by={K}}] ;
    \node[circle,minimum size=2pt,fill=red] at(K) {};
\end{tikzpicture}

\end{document}