Denote the length of the third side of this isosceles triangle as `b . ` Then, the given condition "*the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side*" means that

`1) sqrt ( a ) + sqrt ( a ) = sqrt ( b ) ,`

`2) sqrt ( a ) + sqrt ( b ) = sqrt ( a ) .`

The second condition alone implies `sqrt ( b ) = 0 , ` so `b = 0 . ` If some side of a triangle has zero length, the area of the triangle is zero.

But even the first condition alone leads to the same conclusion. Indeed, it means `2 sqrt ( a ) = sqrt ( b ) , ` which gives `4 a = b ` when squared. But a triangle cannot have sides `a , ` `a ` and `4 a ` because it violates the triangle inequality: `a + a lt= 4 a .` The only option for this to produce at least a degenerate triangle is to have `a = 0 . ` Then `b = 0 , ` too, and the area is zero.

Actually, because of the triangle inequality a condition `a^p + b^p = c^p ` can produce a valid triangle only for `p gt 1 , ` but here `p = 1 / 2 lt 1 .`