# 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. 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 $0$, 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. Then

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.