A Maimed Cake: What is this about?
A Mathematical Droodle

Explanation

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First, introduce some notations. Let the original rectangle be ABCD, with a small rectangle UZRD removed off the corner D, Z being internal to ABCD. The problem is to draw a line that cuts the L-shape ABCRZDU into to pieces of equal area. The solution is obviously not unique. The one below has been motivated by Euclid I.43.

Had Z lied on the diagonal BD, the problem would have been solved with the cut BZ. We shall consider BZ as the first approximation even where Z is away from BD. In the latter case BZ still solves the problem for a different shape cake. Extend BZ to the intersection with either CD or AD, as the case may be. Assume BZ intersects, say, CD first. Through the point of intersection M draw a line parallel to AD. The latter intersects AB in K and ZU in L. BZ divides the L-shape KBCMZL into equal areas. However, the rectangle AKLU is left over. One solution is to upgrade the cut BZ as to append to the area BCMZ a shape of area half that of AKLU. What comes to mind is the triangle BZQ, with ZQ = LU. The triangle has the same base (LU) and height (KL) as the rectangle AKLU and, therefore, half the area of the latter. Hence BQ provides a solution to our problem.

Another solution with a cut PQ parallel to BZ can be obtained by taking BP = ZQ = LU/2. Finally note that the problem can be easily generalized to a rare case of a cake in the shape of parallelogram. As, the only thing that is important to the above solution is that the toddler cuts off another parallelogram UZRD whose sides will naturally be parallel to those of ABCD.

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