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locus of midpoints of arcs of equal length in equal circles, 1 fixed point each

Source: Spanish Mathematical Olympiad 1984 P5

August 2, 2018

geometryLocusarc

Problem Statement

Let

A

and

A'

be fixed points on two equal circles in the plane and let

AB

and

A' B'

be arcs of these circles of the same length

x

. Find the locus of the midpoint of segment

BB'

when

x

varies: (a) if the arcs have the same direction, (b) if the arcs have opposite directions.