The projections {H_{1}, H_{2}} 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 H_{0} of AH and the middle E of BC define a diameter of the Euler circle parallel to OA, O being the circumcenter. Thus triangles AH_{0}H_{1}, H_{1}EF, H_{0}AH_{2} are isosceli etc....

By the way, note that (i) HH_{1}AH_{2} is similar to AFA''A', (ii) Triangle H_{2}EA' is isosceles, (iii) Angle HAF = angle(DEA')/2 = (B-C)/2. (iv) Triangles AH_{0}H_{1}, H_{1}EF, AOF are similar isosceli.