Goormaghtigh's theorem

Goormaghtigh theorem is the name given variously to one of the geometry problems proposed by the French engineer mathematician René Goormaghtigh, individually known as Goormaghtigh’s generalization of Musselman’s theorem and Goormaghtigh’s generalization of Droz-Farny line theorem.

Goormaghtigh’s generalization of Musselman’s theorem

This theorem states that let $ABC$ be a triangle, $O$ be the circumcircle of $ABC$. If $A',B',C'$ lie on $OA,OB,OC$ such that ${\frac {OA'}{OA}}={\frac {OB'}{OB}}={\frac {OC'}{OC}}$ then the intersections of the perpendiculars to $OA$ at $A_{0}$, $OB$ at $B_{0}$,and $OC$ at $C_{0}$ with the respective sidelines $BC,CA,AB$ are collinear. We can see detail of the theorem in .

Goormaghtigh’s generalization of Droz-Farny line theorem

This theorem states that given triangle $ABC$ and point $P$ distinct from $A,B,C$. Let a line $\Delta$ passes through $P$. Let $A_{1},B_{1},C_{1}$ belong to $BC,CA,AB$ respectively such that $PA_{1},PB_{1},PC_{1}$ are the images of $PA,PB,PC$ respetively by reflection $R_{\Delta }$. Then, $A_{1},B_{1},C_{1}$ are collinear. Where Notation $R_{\Delta }$ refers to reflection against $\Delta$.

• Special case, when $P$ is the orthocenter of triangle $ABC$, this theorem actually becomes Droz-Farny line theorem

• General case, Dao Thanh Oai's generalization of this theorem in

References

• R. Goormaghtigh, Sur une généralisation du théoreme de Noyer, Droz-Farny et Neuberg, Mathesis 44 (1930) 25.