If two equal sides of an isosceles triangle have length a, find the length of the third side that maximizes the area of the triangle

Consider the attached image. Let's focus on the triangle ACD. This is a right triangle, and is half of ABC. So, if we maximize the area of ACD, we also maximize the area of ABC.

We know that . Let's call the height . We can deduce the other leg:

The area of ACD is given by half the product between the base and the height:

and we want this quantity to be maximized. Again, it is sufficient to maximize the numerator.

To maximize it, let's compute its derivative:

We have a critical point where the derivative equals zero, and a fraction equals zero if and only if its numerator equals zero. So, the critical points are given by

We can't accept a negative solution, so the answer will be

Now, remember that the third side is twice AD, which means

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