EGMO Problem 1.10

By universal_Daniel April 15, 2021

Here’s a really neat problem that demonstrates the usage of lemmas in a geometry proof:

Problem : Show that a trapezoid is cyclic if and only if it is isosceles.

Solution:

–Lemma 1: A trapezoid is cyclic if it is isosceles.

–Proof of Lemma 1: Label the trapezoid as $ABCD$ . Let $BC$ be the longer base of the trapezoid. Since the trapezoid is isosceles, $\angle B = \angle C$ . Because $AD \parallel BC$ , $\angle A + \angle B = 180$ . Therefore, $\angle A = 180^{\circ} - \angle B$ .

Similarly, $\angle D = 180-\angle C$ .

Since $\angle B = \angle C$ , we must have $\angle A = \angle D$ .

We add $\angle A$ and $\angle C$ to get the result: $\angle A + \angle C = 180^{\circ} - \angle B + \angle C = 180^{\circ}$ .

Since the opposite angles of the trapezoid add up to $180^{\circ}$ , the isosceles trapezoid is cyclic.

–Lemma 2: A trapezoid is iscosceles if it is cyclic.

–Proof of Lemma 2: Draw trapezoid $ABCD$ , having $AB$ be the shorter base and $CD$ be the longer base. Draw the circumcircle $\omega$ of the trapezoid.