Problem, aops, https://artofproblemsolving.com/community/c6t48f6h2267253_equal_angles_wanted_trapezoid_equal_segments_right_angle_related

 My solution: 

aops, https://artofproblemsolving.com/community/c4t48f4h3520153_angle_is__always_45

Still using inversion, 

 

Geometria Super Top, Hakan Ersoz, https://www.facebook.com/groups/830782544358783

 A nice an easy to understand synthetic solution to this beautiful problem

FB page of Kenan Rustemov, https://www.facebook.com/photo?fbid=802536598658713&set=pcb.802536625325377

 My solution

aops, https://artofproblemsolving.com/community/c6t48f6h2547832_triangle_with_60ampdeg_angle_projection_of_othocenter_onto_median

My solution

aops, https://artofproblemsolving.com/community/c6t48f6h144817_find_the_angle

 My solution: