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Explanation of the construction: AD = BD, since they are both PQ. Triangle ADB is isosceles, so angle DAB is congruent to angle ABD. Triangle DAC is congruent to triangle DBC by SSS, so angles CDA and CDB are congruent. Then triangles AED and BED are congruent by ASA, so AE = BE and angle AED = angle BED, which makes them right angles since they are a linear pair. Thus line CD is the perpendicular bisector of segment AB.

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