Given its unmatched importance in mathematics and applications, it is wonderful that the Pythagorean theorem (Elements I.47) is apparently not as widely known as one could expect:

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 19^th 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 AB_aC_a, A_bBC_b, A_cB_cC have the same area as ABC [Exercise, p. 736]. To see this, rotate (drag the slider), e.g., AB_aC_a through 90^o clockwise till B_a 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 AM_a 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 ABC_a and an altitude in BCC'_a. It also has been noticed that AM_a = BC/2 and similarly for the other medians.

The triangles AB_aC_a, A_bBC_b, A_cB_cC 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 ABC_a and a median in BCC'_a. We may restate this as follows.

Let, for a triangle center P of ABC, P_a, P_b, and P_c denote its namesakes in triangles AB_aC_a, A_bBC_b, A_cB_cC. Thus, for example, G_a stands for the centroid (the meeting point of the medians) of triangle AB_aC_a. We then have two facts.

Following are additional properties of Vecten's configuration, see [Exercise, pp. 860-861]:

1. Lines AA_b and BB_a meet on the C-altitude of ABC [Exercise, p. 225].

2. Lines AA_b and CC_b are perpendicular. (Assume they cross at point S_b and introduce similarly S_a and S_c.)

3. Line A_cC_a passes through S_b and bisects a pair of angles at that point.

4. Lines AS_a, BS_b, and CS_c are concurrent (and each passes through the center of one of the Vecten's squares.)

5. Let T_a, T_b, and T_c denote the centers of Vecten's squares BCA_cA_b, etc. Then AT_a is equal and perpendicular to T_bT_c.

6. A_cB_c² + B_aC_a² + A_bC_b² = 3(AB² + BC² + AC²) [Exercise, p. 736].

Ernst Wilhelm Grebe, was the first to call the common point of AT_a, BT_b, and CT_c 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 O_aO_bO_c 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 P_aP_bP_c at Q. Because of the symmetry of the flank relationship, friendship is also symmetric: ABC and P_aP_bP_c are perspective at Q, then ABC and Q_aQ_bQ_c 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 A_bB_a does not depend on C. Furthermore, he proved that triangles AMB and A_cMB_c 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 A_cB_c inwardly to the flank A_cB_cC. CM is the C-cevian in ABC and A_cB_cC 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 K(p/2 - f) are related by friendship. In particular, the Fermat points K(±p/3) are friends with respective Napoleon's points K(±p/6).

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.

(Further results could be found at the already quoted paper by F. van Lamoen and the two online articles by P. Yiu.)

References

1. F. G.-M., Exercise de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991
2. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.
3. C. Pritchard, General Introduction, in The Changing Shape of Geometry, edited by C. Pritchard, Cambridge University Press, 2003
4. R. Smullyan, 5000 B.C. and Other Philosophical Fantasies, St. Martin's Press, 1983
5. F. van Lamoen, Friendship Among Triangle Centers, Forum Geometricorum 1 (2001), pp. 1-6.
6. R. Webster, Bride's Chair Revisited, Mathematical Gazette 78 (November 1994), pp. 345-346. (Reprinted in Pritchard, pp. 246-247.)
7. I. Warburton, Bride's Chair Revisited Again!, Mathematical Gazette 80 (November 1996), pp. 557-558. (Reprinted in Pritchard, pp. 248-250.)
8. P. Yiu, Squares Erected on the Sides of a Triangle
9. P. Yiu, On the Squares Erected Externally on the Sides of a Triangle

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 7,000,000^th visitor. Most recently the site has been recognized with the 2003 Sci/Tech Award from the editors of Scientific American.

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