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The diameter of a circle is a circle's largest chord.
Yes, a cord that is a diameter always passes through the center of a circle. And it's also the longest cord of that circle.
Yes, that is the definition of a semicircle.
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.
The base of a cone is always a circle. it also can be a ellipse.