Circle and its diameter are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on and two other vertices on
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Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
Izvor: Međunarodna matematička olimpijada, shortlist 1967
Školjka 
and its diameter 
are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on 