also that triangles AGF and AHI are also similar in the attached figure.
As lines HK and GD are parallel (by construction), angles HKA and GDA are
congruent. Also angle DAG is common to both triangles all of the angles in
AGD are congruent to the corresponding angles in AHK, and so these
triangles are similar.
Now look at angles HIA and GFA, these are both right angles (by
construction), and angle HAI is common to both triangles AGF and AHI
so these two triangles have two equal angles, hence all the corresponding
angles are congruent, and so the triangles are similar.