Musselman's theorem

The theorem was proposed as an advanced problem by John Rogers Musselman and René Goormaghtigh in 1939,[4] and a proof was presented by them in 1941.[5] A generalization of this result was stated and proved by Goormaghtigh.[6]

The generalization of Musselman's theorem by Goormaghtigh does not mention the circles explicitly.

As before, let ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ be the vertices of a triangle ${\displaystyle T}$, and ${\displaystyle O}$ its circumcenter. Let ${\displaystyle H}$ be the orthocenter of ${\displaystyle T}$, that is, the intersection of its three altitude lines. Let ${\displaystyle A'}$, ${\displaystyle B'}$, and ${\displaystyle C'}$ be three points on the segments ${\displaystyle OA}$, ${\displaystyle OB}$, and ${\displaystyle OC}$, such that ${\displaystyle OA'/OA=OB'/OB=OC'/OC=t}$. Consider the three lines ${\displaystyle L_{A}}$, ${\displaystyle L_{B}}$, and ${\displaystyle L_{C}}$, perpendicular to ${\displaystyle OA}$, ${\displaystyle OB}$, and ${\displaystyle OC}$ though the points ${\displaystyle A'}$, ${\displaystyle B'}$, and ${\displaystyle C'}$, respectively. Let ${\displaystyle P_{A}}$, ${\displaystyle P_{B}}$, and ${\displaystyle P_{C}}$ be the intersections of these perpendicular with the lines ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$, respectively.

It had been observed by Joseph Neuberg, in 1884, that the three points ${\displaystyle P_{A}}$, ${\displaystyle P_{B}}$, and ${\displaystyle P_{C}}$ lie on a common line ${\displaystyle R}$.[7] Let ${\displaystyle N}$ be the projection of the circumcenter ${\displaystyle O}$ on the line ${\displaystyle R}$, and ${\displaystyle N'}$ the point on ${\displaystyle ON}$ such that ${\displaystyle ON'/ON=t}$. Goormaghtigh proved that ${\displaystyle N'}$ is the inverse with respect to the circumcircle of ${\displaystyle T}$ of the isogonal conjugate of the point ${\displaystyle Q}$ on the Euler line ${\displaystyle OH}$, such that ${\displaystyle QH/QO=2t}$.[8][9]