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. According to D. E. Smith [Smith, p. 289], the Arabs call it the Figure of the Bride. He also mentions [Smith, p. 290], with a reference to E. Lucas' Récréations Mathématiques, that the Greeks call it the theorem of the married women, while Bhãskara is said to have spoken of it as the chase of the little married women.

(Florian Cajori ascribes (Am Math Monthly, f 6, no 3, 1899, p. 72) the source of the Bride's Chair designation to a mistranslation of the Greek word numfh "applied to the theorem by a Byzantine writer of the 13 century. This Greek word admits of two meanings, 'bride' and 'winged insect.' The figure of the right triangle with the three squares suggests an insect but Behâ Eddîn apparently translated the word as 'bride'." Probably out of respect of the famous mathematician and his theorem.)

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.

A different kind of generalization kept the squares on the sides with a relaxed requirement on the angles of the triangle. Euclid himself considered an arbitrary triangle with squares on its sides in Prop. 63 of his surviving work The Data. In the 19^th century (1817), Vecten also studied arbitrary triangles 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 ΔABC [Exercices, p. 736, Five Hundred, #295]. To see this, rotate (drag the slider), e.g., ΔAB_aC_a through 90^o clockwise till B_a coincides with C. Then CA will serve as a median of ΔBCC'_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 ΔBCC'_a, parallel to its side BC, which implies the second property, viz., the same line serves as a median in ΔABC_a and an altitude in ΔBCC'_a. It also has been observed 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 ΔABC. 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 ΔAB_aC_a and a median in ΔABC. We may restate this as follows.

Let, for a triangle center P of ΔABC, 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 [Exercices, pp. 860-861]:

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

2. Lines AA_b and CC_b are perpendicular and equal. (This follows from the congruence of triangles ABA_b and C_bBC.) Assume AA_b and CC_b meet 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. (Quadrilateral AC_aC_bS_b is cyclic as having opposite angles at C_a and S_b both 90°. In the circumcircle, angles AS_bC_a and AC_bC_a, the latter being 45°, are subtended by the same chord AC_a. Hence angle AS_bC_a is also 45°.)

4. Lines AS_a, BS_b, and CS_c are concurrent and each passes through the center of one of 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) [Exercices, 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 ΔABC.

Further results were obtained by Ernst Wilhelm Grebe in 1847 and expanded by F. van Lamoen in 2001.

Grebe has shown [Exercices, p. 1181] that if the outer sides of Vecten's squares have been extended to form ΔA'B'C', then the latter is similar, in fact homothetic, to ΔABC. 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 ΔABC, it is also called the symmedian point K. For K the distances to the sides of ΔABC are proportional to the sides themselves [Honsberger, p. 59], 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 ΔABC at K. The points O and K thus stand in a certain relationship, which F. van Lamoen termed friendship.

In general, centers P and Q are friends if ΔABC is perspective to ΔP_aP_bP_c at Q. Because of the symmetry of the flank relationship, friendship is also symmetric: if ΔABC and ΔP_aP_bP_c are perspective at Q, then ΔABC and ΔQ_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 ΔABC and of another, constructed on A_cB_c inwardly to the flank A_cB_cC. CM is the C-cevian in ΔABC and ΔA_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 ΔABC. 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).

(Further results could be found in the already quoted paper by F. van Lamoen and at this site .)

References

1. E. J. Barbeau, M. S. Klamkin, W. O. J. Moser, Five Hundred Mathematical Challenges, MAA, 1995.
2. C. M. Taisbak, Euclid's Data: The Importance of Being Given, Museum Tusculanum (June 2003).
3. F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991
4. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.
5. C. Pritchard, General Introduction, in The Changing Shape of Geometry, edited by C. Pritchard, Cambridge University Press, 2003
6. D. E. Smith, History of Mathematics, v. II, Dover, 1958
7. R. Smullyan, 5000 B.C. and Other Philosophical Fantasies, St. Martin's Press, 1983
8. F. van Lamoen, Friendship Among Triangle Centers, Forum Geometricorum 1 (2001), pp. 1-6.
9. R. Webster, Bride's Chair Revisited, Mathematical Gazette 78 (November 1994), pp. 345-346. (Reprinted in Pritchard, pp. 246-247.)
10. I. Warburton, Bride's Chair Revisited Again!, Mathematical Gazette 80 (November 1996), pp. 557-558. (Reprinted in Pritchard, pp. 248-250.)

Copyright © 1996-2008 Alexander Bogomolny