Cfrgtkky

Following this answer, would it also be possible to have similar styles to find the intersection point(s) for cases where the circle is defined in terms of (1) three points; (2) a point and a center?

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}

(For example, (1) the intersection point of the circle passing through A,B,C and the line segment D-E; (2) the intersection point of the circle passing through A with center B and the line segment D-E.)

For case (1): Based on this nice answer,

For case (2): Using simply through library.

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}