Question

A circle is circumscribed about a trapezoid. Prove that this is possible if and only if the trapezoid is isosceles.

Answer 1

Proof

Let ABCD be an isosceles trapezoid with the bases AB and CD and the lateral sides AD and BC (Figure 1a). We need to prove that there is a circle which passes through all the vertices of the trapezoid A, B, C and D. Let us draw the diagonals of the trapezoid AC and BD (Figure 1b) and consider the triangles ABC and ABD. These triangles have the common side AB and the congruent sides BC and AD (the latest is because the trapezoid ABCD is isosceles).

Figure 1a. To the Problem 1

Figure 1b. To the solution of the Problem 1

The angles BAD and ABC concluded between these congruent sides are congruent as the base angles of the isosceles trapezoid.
Hence, the triangles ABC and ABD are congruent in accordance with the SAS-test for the triangles congruency.
It implies that the angles ACB and ADB are congruent as the corresponding angles of congruent triangles.
Thus the angles ACB and ADB are congruent and are leaning on the same segment AB. Hence, these angles are inscribed in a circle.
The converse statement is true that if the trapezoid is inscribed in a circle, then the trapezoid is isosceles.
By combining the direct and the converse statements you can conclude that a trapezoid can be inscribed in a circle if and only if the trapezoid is isosceles.