Given an acute triangle $ABC$, $d$ is its Euler line. $M, N$ is the reflection of $B, C$ via $d$. $P$ is an arbitrary point on $d$.
$PM$ intersects $AC$ at $E$, $PN$ intersects $AB$ at $F$. $S$ is the reflection of $H$-the orthocenter of triangle $ABC$ via $EF$. Prove that $S$ lies on the circumcircle of triangle $ABC$.
Prove that $PS$ passes through a fixed point.
Solution: Let $(O)$ be the circumcircle of $\triangle ABC$ and $L$ the reflection of $A$ on $OH.$ $d \equiv OH$ cuts $BC,CA,AB$ at $A',B',C'.$ When $P$ varies on $d,$ then $MP \mapsto NP$ is a perspectivity inducing a homography $E \mapsto F$ between $AC$ and $AB$ $\Longrightarrow$ $EF$ envelopes a fixed conic that touches $AC,AB.$ When $P$ coincides with $A',B',C',$ then $EF$ coincides with the reflections $MN,NL,LM$ of $BC,CA,AB$ on $OH,$ respectively, all tangent to the inconic $\mathcal{C}$ with foci $O,H$ by obvious symmetry $\Longrightarrow$ all $EF$ touch $\mathcal{C}$ $\Longrightarrow$ projection $T$ of $H$ on $EF$ is on pedal circle of $\mathcal{C};$ the 9-point circle $(N_9)$ of $\triangle ABC$ $\Longrightarrow$ reflection $S$ of $H$ about $T$ is then on the image $(O)$ of $(N_9)$ under homothethy with center $H$ and coefficient 2.
Let $EF$ cut $BC$ at $D$ and let $AH$ cut $(O)$ again at $U.$ Since $BC$ and $EF$ are perpendicular bisectors of $\overline{HU}$ and $\overline{HS},$ then $D$ is the circumcenter of $\triangle HUS$ $\Longrightarrow$ $OD \perp US$ is perpendicular bisector of $\overline{US},$ hence the application $OD \mapsto US$ is homographic, but as $D \mapsto E$ is a homography between $BC,CA$ and $E \mapsto P$ a perspectivity between $CA,d,$ we get $S \ \overline{\wedge} \ P$ with double points at $\{d \cap (O) \}$ $\Longrightarrow$ $S \mapsto P$ is a stereographic projection of $(O)$ onto $d$ $\Longrightarrow$ all $PS$ go through a fixed point $X \in (O).$ Observing the cases where $P$ coincides with $A',B'$ or $C',$ bearing in mind that $H$ is also orthocenter of $\triangle MNL,$ we deduce that this fixed $X$ is the Euler's reflection point of $\triangle MNL.$
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