# An isosceles trapezoid is described around the circle. The angle for a larger base is 60 degrees.

**An isosceles trapezoid is described around the circle. The angle for a larger base is 60 degrees. What is the ratio of the lengths of the bases?**

According to the properties of a trapezoid circumscribed around a circle, the sum of the lengths of its lateral sides is equal to the sum of the lengths of its bases.

AB + CD = BC + AD.

Let’s draw the height of the trapezoid ВН. In a right-angled triangle ABН, the leg AH lies opposite the angle 30 and is equal to half the length of the hypotenuse AB. AH = AB / 2.

Since the trapezoid is isosceles, then DK = AB / 2.

Base AD = AH + НK + DK = AB / 2 + BC + AB / 2 = AB + BC.

By the property of an isosceles trapezoid circumscribed around a circle, the lateral side of the trapezium is equal to the midline of the trapezoid.

AB = (BC + AD) / 2.

AD = ((BC + AD) / 2) + BC.

AD = (BC + AD + 2 * BC) / 2.

AD * 2 = 3 * BC + AD.

AD = 3 * BC.

ВС / АD = 1/3.

Answer: The ratio of the bases of the trapezoid is 1/3.