no
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Not always but sometimes
yes! But only an isosceles Trapezoid.
If you want to, you can always draw a circle around an isosceles trapezoid and the radius can be half the base of the trapezoid.
yes
An isosceles triangle has at least two congruent sides. An equilateral triangle has three congruent sides. So, an equilateral triangle is a special case of isosceles triangles. Since the equilateral triangle has three congruent sides, it satisfies the conditions of isosceles triangle. So, equilateral triangles are always isosceles triangles. Source: www.icoachmath.com