Dependency of locus sets on the initial construction 1

Given a triangle ABC and a free point D we consider the orthogonal projections E, F and G of D onto the sides of the triangle. For what positions of D are points E and F equidistant to G? We obtain the answer by typing LocusEquation[h==i,D] where h and i are the (lengths of the) segments joining F, G and E, G respectively.
The correct positions of D are shown with green. Also the segments h and i are shown with the same color.

What types of loci can be obtained for those points D for which h=i? By dragging A, B and C, try to find the answer.

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