Sections of triangles and circles
Consider any triangle \(ABC\). Let \(G_A\) be any point on the median line \(s_A\) through \(A\) and midpoint \(M\) of \(BC\), excepting \(A\) and \(M\). Denoting \(E\) (resp. \(F\)) the intersection point of the lines connecting \(G_A\) with \(B\) (resp. \(C\)) and \(A\) with \(C\) (resp. \(B\)), call \(A\)-triangles the four triangles \(ABM\), \(AMC\), \(ABE\) and \(ACF\). Then the following three properties are shown. The centers of the circumscribed circle of each of the four \(A\)-triangles lie on a circle, which we call an \(A\)-circle. The locus of all centers of \(A\)-circles when \(G_A\) moves along \(s_A\) is a straight line parallel to \(s_A\), which we call the \(A\)-line. The \(A\)-line, \(B\)-line and \(C\)-line are concurrent.
scientific article; zbMATH DE number 7633071