19-Reflection Fermat points in circumcircle and Nine points circle

Let  be a triangle,  are first and secon Fermat points. Denote:

 respectively are reflection of First and secon Fermat point in Nine point circle

 respectively are reflection of First and secon Fermat point in Circumcircle (ABC)

Then six points:  lie on a circle

Solution by Telv Cohl:

Lemma: $ F_+, F_- $ is the image of each other under the inversion WRT the Orthocentroidal circle of $ \triangle ABC $ .

Proof of the lemma :

Let $ \triangle N_aN_bN_c $ be the First Napoleon triangle of $ \triangle ABC $ .

Let $ \triangle N_a'N_b'N_c' $ be the Second Napoleon triangle of $ \triangle ABC $ .

Let $ G, H $ be the Centroid, Orthocenter of $ \triangle ABC $ , respectively .

Let $ T $ be the intersection of $ F_+F_- $ and the line passing through $ G $ and tangent to $ (GF_+F_-) $ .

It's well known that $ F_+, N_a', N_b', N_c' $ are concyclic and $ F_-, N_a, N_b, N_c $ are concyclic .

From this problem Another problem (for $ \triangle GN_aN_a'$ and $ \triangle GN_bN_b' $ and $\triangle GN_cN_c' $ ) we get the reflection of $ T $ in $ G $ is $ N_aN_a' \cap N_bN_b' \cap N_cN_c' \equiv O $ , so we get $ T $ is the midpoint of $ GH $ and $ TF_+*TF_-=TG^2=TH^2 $ . ie. $ F_+, F_- $ is the image of each other under the inversion WRT $ (GH) $

Back to the main problem :

I'll only prove the case when $ (N) $ and $ (O) $ doesn't have common points 

( and we can use similar discussion to prove other case )

Let $ L, L' $ be the limit points of the coaxial system determined by $ \{ (N) , (O) \} $ .

Since $ G , H $ is the insimicenter, exsimicenter of $ (N) $ and $ (O) $ , respectively .

so $ (GH) , (N)  , (O) $ are coaxial , 

hence $ L, L' $ is the image of each other under the inversion WRT $ (GH) $ .

From the lemma we get $ F_+, F_- $ is the image of each other under the inversion WRT $ (GH) $ ,

so we get $ L, L', F_+, F_- $ are concyclic at $ \omega $ . 

Since $ L, L' $ lie on $ \omega $ ,

so $ \omega $ is orthogonal to $ (O) $ and $ (N) $ ,

hence $ F_{+N}, F_{-N}, F_{+O}, F_{-O} $ all lie on $ \omega $ .

ie. $ F_+, F_-, F_{+N}, F_{-N}, F_{+O}, F_{-O} $  are concyclic 

Q.E.D