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Maximizing the incircle area in an isosceles triangle

Maximizing the area of the Inscribed circle in a isosceles triangle of leg length 1

Drag the base angle slider to change the base angle of the isosceles triangle, or hit the start button to animate. If the length of each leg is 1, the area of the incircle, which is tangent to all three sides, is maximal when the base length is sqrt(5) - 1 and the base angle is the inverse cosine of ((sqrt(5) - 1) / 2). Challenge problem: use calculus to prove the above statement. It will be very helpful to use the fact that the area of the triangle equals one-half times the radius of the incircle times the perimeter of the triangle, along with Heron's formula.