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Isosceles triangle is inscribed in a circle.

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$\"\"$

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$\"\"$ be the radius of the circle.

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$\"\"$ be the base of the triangle.

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Observe the figure.

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From the figure, height of the triangle is $\"\"$ .

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From the pythagorean theorem,

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$\"\"$

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The area of the triangle is $\"\"$ .

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$\"\"$

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The area of the triangle is maximum when its first derivative becomes zero.

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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Equate $\"\"$ to zero.

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$\"\"$

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The general solution for a quadratic equation is $\"\"$ .

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$\"\"$

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Here $\"\"$ can not benegative.

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Hence $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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Therefore the height of the triangle is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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Therefore the base of the triangle is $\"\"$ .

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The height of the triangle is $\"\"$ .

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The base of the triangle is $\"\"$ .