Equicevian points on the altitudes of a triangle (Q1941571)

Let \(ABC\) be a triangle. Consider a point \(P\neq B\) a point in the plane containing \(ABC\) such that \(BP\) is not parallel to \(AC\). We denote (it it exists) by \(BB_P\) the cevian from \(B\) through \(P\). In the same way, if \(P\neq C\) and \(CP\) is not parallel to \(AB\), we could define \(CC_P\). In this situation a point \(P\) is called \(A\)-\textit{equicevian} if the lengths of \(BB_P\) and \(CC_P\) are equal. The paper under review is concerned to \(A\)-equicevian points lying on the altitude \(AO\) from \(A\). In passing, some remarks concerning the (somewhat ubiquitous) polynomials \(p(X,Y,Z)=X^3+Y^3+Z^3-3XYZ\) and \(q(T)=T^3-(\alpha^2-\beta^2-\gamma^2)T+2\alpha\beta\gamma\) are made.