Grebe: from Ladies' Diary to Carroll's Pillow

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Grebe: from Ladies' Diary to Carroll's Pillow

The following problem has been proposed in the 1952 (November) issue of Mathematics Magazine, with four solutions published in Vol. 26, No. 5 (May - Jun., 1953), pp. 279-282. One of the solutions (by Leon Bankoff) is illustrated below.

Upon the sides of triangle ABC the squares ABDE, BCFG, ACHL are constructed extrerior to the triangle. Construct triangle ABC given A', B', C' which are the intersections of DE and HL, ED and FG, GF and LH, respectively.

The problem was accompanied by an editor's remark:

The problem itself has always aroused interest. Five solvers successfully attacked it for the right triangle in Question 1615 of the Ladies' Diary of 1838. Charles L. Dodgson gives a geometric and a trigonometric solution as Problem 57 in Pillow Problems, 3rd Edition, Macmillan, London, 1894. Most modern geometries deal with it or its essentials, for example, Altshiller-Court, College Geometry,Johnson (1925), page 232.

It was also noted by the proposer (D. L. MacKay) that the problem is based on the theorem regarding the concurrency of AA', BB', CC' given by E. W. Grebe in 1847 and that, as such, it is the basis of the German claim that the symmedian point should be named Grebe's point.

(Charles L. Dodgson is better known by his literary pseudonym of Lewis Carroll.)

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L. Bankoff's solution makes use of Grebe's theorem.

Start with drawing squares on the sides of triangle A'B'C' and extend their outer sides to form triangle A₀B₀C₀. By Grebe's theorem triangles A'B'C' and A₀B₀C₀ are homothetic from K, the common Lemoine point of the two triangles. Since the same is true for triangles ABC and A'B'C', all three are homothetic from K. Join the far corners of the squares on the sides of ΔA'B'C' to K. And mark the points of intersection with the sides of that triangle. (My notations confirm to those on Grebe's theorem page.) Because of the established homothety of ABC and A₀B₀C₀ this points will serve as the far corners of the squares on the side of the sought triangle ABC thus defining the latter unambiguously.

The Ladies' Diary was a popular publication that appeared annually from 1704 through 1831 in London. The magazine was "designed for the use and diversion of the fair sex." Among calendar material, important dates, etc., it also contained a puzzle and problem section. The section grew with time and, on occasion, included quite serious problems.

Ivars Peterson addressed the publication in one of his early MAA Online columns where he quoted several sample puzzles from The Ladies' Diary. Here's one from the 1809 volume:

I'm a singular creature, pray tell me my name,
I partake of an Englishman's freedom and fame;
I daily am old, and I daily am new,
I am praised, I am blam'd, I am false, I am true;
I'm the talk of the nation, while still in my prime,
But forgotten when once I've outlasted my time.
In the morning no Miss is more coveted than I,
In the evening no toy thrown more carelessly by.
Take warning, ye fair! I, like you, have my day,
And, alas! You, like me, must grow old and decay.

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