Once in a kingdom far, far away, a beautiful girl was born to a king & queen. The castle in which she was born was a structure to behold. The diagonals of the side walls divide each other into segments with lengths that are pairwise equal; in terms of the picture below,
AE =
DE,
BE =
CE (and
AE ≠
CE if one wishes to exclude rectangles). Opposite angles are supplementary, which in turn implies that isosceles trapezoids are cyclic quadrilaterals, at least according to the fairy godmother who helped construct the magical castle.
But a jealous witch wanted to destroy the kingdom, and she knew too well that the castle would only fall if the ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is,
The length of each diagonal is, according to Ptolemy's theorem, given by
where
a and
b are the lengths of the parallel sides
AD and
BC, and
c is the length of each leg
AB and
CD.
But luckily the witch miscalculated her assault, and the young princess & her kingdom lived happily ever after.