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.)