Problem — Isosceles trapezoid from parallel chords

Text

In a circle, two parallel chords $PQ$ and $RS$ are drawn. Prove that their four endpoints are the vertices of an isosceles trapezoid (or of a rectangle, in the special case where the two chords are congruent).

Solution

Order the vertices so that the quadrilateral is $PQSR$, with the sides $PQ$ and $RS$ parallel by hypothesis: it is therefore a trapezoid, with legs $PR$ and $QS$. Since the chords $PQ$ and $RS$ are parallel, the arcs $PR$ and $QS$ lying between them are congruent (arcs between parallel chords). Congruent arcs correspond to congruent chords, so $\overline{PR} = \overline{QS}$: the legs are congruent and the trapezoid is isosceles. If in particular $\overline{PQ} = \overline{RS}$, the trapezoid has congruent bases and is therefore a parallelogram; but a parallelogram inscribed in a circle is a rectangle. $\blacksquare$