Solution to "Scarecrow's Theorem"

No, Scarecrow's Theorem is in fact never true. So it's not so much a theorem as it is a... lie.

Take some isosceles triangle with sides a, a, b, and angle \theta between the two equal sides as in the image.

isosceles triangle

Now, according to Scarecrow's Theorem, there are some a, b, and \theta such that either (1) \sqrt{a}+\sqrt{b}=\sqrt{a}, or (2) \sqrt{a}+\sqrt{a}=\sqrt{b}. For case (1) to be true, b would have to equal zero, and therefore there would be no triangle so that doesn't work. For case (2), let's look at some extreme values for b in comparison with a. As \theta approaches 0, b approaches 0,
isosceles triangle

so the minimum value for b is just above 0. Now, as \theta approaches \pi, then b maxes at just shorter than 2a, since if b=2a then it would just be a straight line, no triangle.
isosceles triangle



But if Scarecrow's Theorem were true for case (2) then




But b cannot equal 4a since we have just shown b must be between 0 and 2a, therefore Scarecrow's Theorem is not true for all cases, and thus is never true.