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 , , , and angle between the two equal sides as in the image.
Now, according to Scarecrow's Theorem, there are some , , and such that either (1) , or (2) For case (1) to be true, 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 in comparison with . As approaches , approaches ,
so the minimum value for is just above . Now, as approaches , then maxes at just shorter than , since if then it would just be a straight line, no triangle.
But if Scarecrow's Theorem were true for case (2) then
But cannot equal since we have just shown must be between and , therefore Scarecrow's Theorem is not true for all cases, and thus is never true.