The projections {H1, H2} of the orthocenter H on the two bisectors at A of triangle ABC are on the line joining the middle E of BC with the center D of the Euler circle.
The middle H0 of AH and the middle E of BC define a diameter of the Euler circle parallel to OA, O being the circumcenter. Thus triangles AH0H1, H1EF, H0AH2 are isosceli etc....
By the way, note that (i) HH1AH2 is similar to AFA''A', (ii) Triangle H2EA' is isosceles, (iii) Angle HAF = angle(DEA')/2 = (B-C)/2. (iv) Triangles AH0H1, H1EF, AOF are similar isosceli.
![[alogo]](alogo.png){kind=small}
Orthocenter projections ![[0_0]](OrthocenterProjections_files/OrthocenterProjections_0_0_0.png)
![[0_1]](OrthocenterProjections_files/OrthocenterProjections_0_0_1.png)
![[1_0]](OrthocenterProjections_files/OrthocenterProjections_0_1_0.png)
![[1_1]](OrthocenterProjections_files/OrthocenterProjections_0_1_1.png)