Gossard's perspector and projective consequences (Q2871652)

A fairly well known theorem of Euler states that the centroid \(\mathcal{G}\), the circumcenter \(\mathcal{O}\), and the orthocenter \(\mathcal{H}\) of a non-equilateral triangle are collinear, forming what is now known as the Euler line, and that \(\|\mathcal{O} \mathcal{G}\| : \| \mathcal{O} \mathcal{H} \| = 1 : 3\). Less well known is the fact, also proved by Euler, that the Euler line \(L_A\) of the triangle formed by the Euler line of \(ABC\) and the two sides \(AB\) and \(AC\) is parallel to \(BC\), with similar definitions of and statements about \(L_B\) and \(L_C\). A related result, proved by Gossard in 1915 states that the triangle formed by \(L_A\), \(L_B\), and \(L_C\) is triply perspective with \(ABC\) and has the same Euler line as \(ABC\). Another related property of the Euler lines appeared in the first problem of the 1997 W. L. Putnam competition, where contestants were asked to prove that if one of four lines in most general position is parallel to the Euler line of the triangle formed by the remaining three, then each one of the four lines is parallel to the Euler line of the triangle formed by the remaining three.NEWLINENEWLINEAmong other things, the paper under review gives new and more elementary proofs of the results mentioned above, and explores the deeper geometric meaning of a phenomenon seen in them. It explores from a projective viewpoint the geometric structure inspired by Euler's results, the relative position of the Euler line and the three sidelines of the triangle, and relation to the nine-point circle. The paper relates Pappus' theorem to the nine-point circle and the Euler line.