Problem
GEO-B1-M04-P023 A Cyclic Parallelogram
#23
★★★★☆ Level 4 of 5
Parallelogram \(ABCD\) is cyclic: its vertices lie on one circle. Prove that \(ABCD\) is a rectangle.
In a parallelogram, opposite angles are equal; in a cyclic quadrilateral, they sum to \(180^\circ\).
In a parallelogram, \(\angle A=\angle C\). Since the quadrilateral is cyclic, opposite angles are supplementary, so \(\angle A+\angle C=180^\circ\). But these angles are equal, hence each is \(90^\circ\). Similarly, the remaining angles are also right. Therefore \(ABCD\) is a rectangle.
A strong combination of two topics: parallelogram plus circle.