Chords, loci and the four steps of a construction
See that the segment from the centre to the midpoint of a chord is perpendicular to it, find the distance to a chord and count the solutions of a locus construction.
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الدليل
Ukrainian Geometry 7 (§§ 21, 24) teaches these: properties of the circle, the tangent, and the locus method in construction problems.
A chord joins two points of a circle. A diameter is a chord through the centre. A diameter perpendicular to a chord bisects it; and conversely, the line from the centre to the midpoint of a chord is perpendicular to it.
Equal chords of a circle are equally far from the centre, and the other way round. The distance from the centre to a chord is a perpendicular: with the radius and half the chord it makes a right triangle.
A locus is the figure made of all points with a property. For example the locus of points at distance r from O is a circle. Points with two properties lie where two loci cross.
A construction problem has four steps: analysis, construction, proof, investigation. In the last one you find how many solutions there are.