Prove that the locus of the midpoints of the parallel chords of a circle is the diameter of the given circle.
2026-05-16 22:47:27.1778971647
A geometry problem on locus
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It is not "the" but merely "a" diameter of the circle, and this follows from a basic theorem of circles:
Theorem: A line segment through a circle's center bisects a cord in the circle iff it is perpendicular to it.
Take now one cord in the circle and the circle's diameter parallel to it (why does such a diameter exist?). If you join both middle points of these two cords then you get a line segment through the circle's center (obvious...why?), and if you continue this line segment to both sides of both cords until it touches the circle at two points, then any cord of the circle parallel to the both above must be interssected in its middle point (i.e. bisected) by this line (since corresponding angles must be equal, and one of this we already know is a straight angle)...