Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
The pictorial representation of the theorem is known in mathematical folklore under many names, the bride's chair, being probably the most popular, but also as the Franciscan's cowl, the peacock's tail and the windmill. In Russia the common name, I believe, is rather pragmatic: Pythagorean pants. In a book of philosophical (!) notes and essays, R. Smullyan tells (pp. 21-22) of an episode in his teaching career. After drawing the diagram, he requested the class to choose the larger of the sum of the small squares and the large square. "Interestingly enough, about half the class opted for the one large square and half for the two small ones. ... Both groups were equally amazed when told that it would make no difference."
Euclid himself generalized the diagram (Elements VI.31) by replacing squares with other shapes. Pappus replaced (Mathematical Collection, Book IV) the squares with arbitrary parallelograms drawn on the sides of an arbitrary triangle.
In the 19th century (1817), Vecten also studied arbitrary triangles but with squares on their sides.
We may immediately observe several simple properties of Vecten's configuration. First of all, the "add-on" triangles ABaCa, AbBCb, AcBcC have the same area as ABC [Exercise, p. 736]. To see this, rotate (drag the slider), e.g., ABaCa through 90o clockwise till Ba coincides with C. Then CA will serve as a median of BCC'a that splits the triangle into two of equal area. (More recently, the fact was observed by R. Webster.) Also, after the rotation, the median AMa will become a midline in BCC'a, parallel to its side BC, which implies the second property, viz., the same line serves as a median in ABCa and an altitude in BCC'a. It also has been noticed that AMa = BC/2 and similarly for the other medians.
The triangles ABaCa, AbBCb, AcBcC are known as flanks of ABC. The relationship is symmetric: a triangle is a flank of its own flanks. Thus, for example, we can also claim that the same line serves as an altitude in ABCa and a median in BCC'a. We may restate this as follows.
Let, for a triangle center P of ABC, Pa, Pb, and Pc denote its namesakes in triangles ABaCa, AbBCb, AcBcC. Thus, for example, Ga stands for the centroid (the meeting point of the medians) of triangle ABaCa. We then have two facts.
where G and H are the centroid and the orthocenter of ABC.
Following are additional properties of Vecten's configuration, see [Exercise, pp. 860-861]:
Ernst Wilhelm Grebe, was the first to call the common point of ATa, BTb, and CTc the Vecten point. There are actually two of them depending on whether the squares have been drawn outwardly (the first Vecten point) or inwardly (the second Vecten point) with regard to ABC.
Further results were obtained by Ernst Wilhelm Grebe in 1847 and expanded by F. van Lamoen in 2001.
Grebe has shown [Exercise, p. 1181] that if the outer sides of Vecten's squares have been extended to form A'B'C', then the latter is similar, in fact homothetic, to ABC. The center of homothety is known as Lemoine's point or (in Germany) as Grebe's point. More neutrally, being the point of intersection of the symmedians in ABC, it is also called the symmedian point K. For K the distances to the sides of ABC are proportional to the sides themselves, and this is the reason for the validity of Grebe's theorem.
For the same reason, another triangle, viz., the triangle OaObOc formed by the circumcenters of the flanks is also homothetic to ABC at K. In general the points O and K stand in a certain relationship, which F. van Lamoen termed friendship.
In general, centers P and Q are friends if ABC is perspective to PaPbPc at Q. Because of the symmetry of the flank relationship, friendship is also symmetric: ABC and PaPbPc are perspective at Q, then ABC and QaQbQc are perspective at P.
What has been shown so far is that O and K are friends, as are G and H (1-2). Quite obviously, the incenter I befriends itself. It's not the only point with that property.
O. Bottema has noticed (see a reference in F. van Lamoen's) that the midpoint M of AbBa does not depend on C. Furthermore, he proved that triangles AMB and AcMBc are both right (at M) and isosceles. Point M therefore serves as the center of the square constructed on AB inwardly to ABC and of another, constructed on AcBc inwardly to the flank AcBcC. CM is the C-cevian in ABC and AcBcC playing the same role in both. We conclude that the second Vecten point that lies at the intersection of such cevians is its own friend.
Similar isosceles triangles on the sides of a given triangle ABC are the subject of Kiepert's theorem that asserts that the outer apexes of the Kiepert triangles form a triangle perspective to ABC. The Kiepert triangles are completely defined by the base angle f (mod p) of the isosceles triangles, and the above perspector is known as the Kiepert perspector K(f). Naturally, the second Vecten point is K(-p/4), where the sign minus indicates that the triangles have been constructed inwardly.
F. van Lamoen's proves a more general fact, viz., that the Kiepert perspectors K(f) and
Also, since the first Vecten point is none other than K(p/4), it follows that the first, like the second, Vecten point befriends itself as well.
A fitting redress for the old diagram.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. In June 2003, his site has welcomed its 7,000,000th visitor. Most recently the site has been recognized with the 2003 Sci/Tech Award from the editors of Scientific American.
Copyright ® 1996-2003 Alexander Bogomolny