Final answer:
A precise proof requires more information about the isosceles trapezoid. However, based on the knowledge that the radius of the inscribed circle in a polygon is related to its area and perimeter, none of the provided answer choices can be determined to be correct without additional context.
Step-by-step explanation:
To prove that the radius of the inscribed circle in the isosceles trapezoid is √xy, we need to use geometric properties and the Pythagorean theorem. The information provided references equations used by the ancient Greeks, and their understanding that the length of a circle with radius r is 2πr. To find the radius of the inscribed circle, we must consider relationships between the trapezoid's dimensions and the circle's radius. Using the Pythagorean theorem (a² + b² = c²), which relates the legs of a right triangle to its hypotenuse, might be particularly helpful.
Based on the options provided (A through D) and without specific dimensions or further context for the isosceles trapezoid, it is not possible to derive the precise formula for the radius of the inscribed circle. In general, the radius of a circle inscribed in a polygon is related to the area and perimeter of the polygon, which is not explicitly given in the question. As per the data given in the question, none of the options directly correlate with a known formula for the radius of an inscribed circle in a trapezoid without additional context or information.