CHAPTER 1: INTRODUCTION TO THE TRIANGLE

1. Isidore Lucien Ducasse, Maldoror (And the Complete Works of the Comte de Lautréamont), translated by Alexis Lykiard (Cambridge, MA: Exact Change, 1994). The original source was Les Chants de Maldoror, published in 1874 and written by the French poet Lautréamont, also known as Comte de Lautréamont, both of which are pseudonyms for Isidore Lucien Ducasse (1846–1870).

2. For more on the Pythagorean theorem, see The Pythagorean Theorem: The Story of Its Power and Beauty, by Alfred S. Posamentier (Amherst, NY: Prometheus Books, 2010).

3. The word obtuse used outside of a mathematical context means dull. Just as an obtuse angle is a dull angle.

4. The word acute used outside of a mathematical context means sharp. Just as an acute angle is a sharp angle.

5. You can find a proof at Wikipedia, s.v. “Apollonius’ theorem,” last modified April 11, 2012, http://en.wikipedia.org/wiki/Apollonius%27_theorem. This relationship is a special case of Stewart's theorem (see chap. 5, p. 121).

6. A proof of this formula was first produced in the book Metrica by Heron of Alexandria in circa 60 CE. More recently, writings of the Arab scholar Abu'l Raihan Muhammed al-Biruni have credited the formula to Heron's predecessor Archimedes prior to 212 BCE (Eric W. Weisstein, “Heron's Formula,” MathWorld, http://mathworld.wolfram.com/HeronsFormula.html).

7. In trigonometric terms, the Pythagorean theorem is sin²∠A + sin²∠B = 1, or sin²α + sin²β = 1.

8. If ∠A = 90°, then cos ∠A = 0, therefore we get a² = b² + c² (theorem of Pythagoras).

9. For more about this ubiquitous ratio, see The Glorious Golden Ratio by A. S. Posamentier and I. Lehmann (Amherst, NY: Prometheus Books, 2012).

10. With some algebraic manipulations we get the following relationships; first, we express c and a in terms of b:

CHAPTER 2: CONCURRENCIES OF A TRIANGLE

1. An altitude of a triangle is the line segment from a vertex perpendicular to the opposite side.

2. An angle bisector of a triangle is the line segment from a vertex to the opposite side and bisecting the angle.

3. A median of a triangle is the line segment joining a vertex with the midpoint of the opposite side.

4. Wilfried Haag, Wege zu geometrischen Sätzen (Stuttgart/Düsseldorf/Leipzig, Germany: Klett, 2003), p. 40.

5. This is a biconditional statement that indicates that if the lines are concurrent then the equation is true, and if the equation is true then the lines are concurrent.

6. Albert Gminder, Ebene Geometrie (München and Berlin: Oldenbourg, 1932), p. 421.

7. The same proof holds true for both an acute and an obtuse triangle.

8. Carl Adams, Die Lehre von den Transversalen in ihrer Anwendung auf die Planimetrie. Eine Erweiterung der euklidischen Geometrie (Winterthur, Switzerland: Druck und Verlag der Steiner'schen Buchhandlung, 1843).

9. A proof by John Rigby can be found in Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry (Washington, DC: Mathematical Association of America, 1995), pp. 63–64.

CHAPTER 4: CONCURRENT CIRCLES OF A TRIANGLE

1. Auguste Miquel, “Mémoire de Géométrie,” Journal de mathématiques pures et appliquées de Liouville 1 (1838): 485–87.

2. The first elementary proof of this relationship was by William Clifford (1845–1879), and the first algebraic proof was published in 2002 by Hongbo Li (“Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra,” Applications of Geometric Algebra in Computer Science and Engineering, edited by L. Dorst et al. (Boston, MA: Birkhauser, 2002), pp. 69–78.

3. R. Johnson, “A Circle Theorem,” American Mathematical Monthly 23 (1916): 161–62.

CHAPTER 5: SPECIAL LINES OF A TRIANGLE

1. In France and England it is called the Lemoine point and in Germany the Grebe point.

2. The point G is here not the centroid of the triangle ABC.

3. The center of gravity of a triangle is the point at which a cardboard triangle can be balanced on the point of a pin.

CHAPTER 6: USEFUL TRIANGLE THEOREMS

1. E.H. Lockwood, A Book of Curves (London: Cambridge University Press, 1971), pp. 76–79, refers to Jakob Steiner, “Über eine besondere Curve dritter Classe (und vierten Grades),” Burchardt's Journal Band LIII, pp. 231–37 (presented at the Academy of Sciences–Berlin, January 7, 1856). Steiner's paper was published in Jacob Steiner, Gesammelte Werke, Band II, edited by K. Weierstrass (Berlin, Germany: G. Reimer, 1882), pp. 639–47.

2. One source for more Simson line relationships can be found in Challenging Problems in Geometry, by A.S. Posamentier and C.T. Salkind (New York: Dover, 1988).

3. Leonhard Euler, “Solutio facilis problematum quorundam geometricorum difficillimorum,” Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 11 (1767): 103–23. Reprinted in Opera Omnia 1, no. 26 (1953): 139–57.

According to the Euler Archive, it was presented to the Petersburg Academy on December 12, 1763 (Euler Archive, “17. All Publciations,” http://www.math.dartmouth.edu/~euler/tour/tour_17.html).

4. Another proof of this collinearity can be found in A.S. Posamentier, Advanced Euclidean Geometry (Hoboken, NJ: John Wiley, 2002), pp. 161–63.

5. C. J. Brianchon and J.-V. Poncelet, “Géométrie des courbes. Recherches sur la détermination d'une hyperbole équilatère, au moyen de quatres conditions donnée,” Annales de Mathématiques pures et appliquées 11 (1820–1821): 205–20.

6. Roger A. Johnson, Advanced Euclidean Geometry (Mineola, NY: Dover, 1960), p. 200.

7. Howard Eves, A Survey of Geometry, rev. ed. (Boston, MA: Allyn & Bacon, 1972; repr. 1965), p. 133.

8. The truly motivated reader might like to see even more points of significance of a triangle. We refer, then, to the website MathWorld, http://mathworld.wolfram.com/KimberlingCenter.html.

CHAPTER 7: AREAS OF AND WITHIN TRIANGLES

1. Jens Carstensen, “Die Seitenhalbierenden—Ein schöner Satz,” Die Wurzel,” July 2004, pp. 160–62.

2. The answer is Area ΔEFG = Area ΔABC.

3. A proof of this can be found at “Napoleon's Theorem,” Mathpages, http://www.mathpages.com/home/kmath270/kmath270.htm.

4. Hugo Steinhaus, Mathematical Snapshots (Mineola, NY: Dover, 1999), p. 9.

5. Ingmar Lehmann, “Dreiecke im Dreieck. Vermutungen und Entdeckungen—DGS als Wundertüte,” Werkzeuge im Geometrieunterricht, edited by Andreas Filler, Mathias Ludwig, and Reinhard Oldenburg (Hildesheim/Berlin, Germany: Franzbecker, 2011), pp. 101–20.

6. Edward Routh (1831–1907) was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge. Edward J. Routh, A Treatise on Analytical Statics, with Numerous Examples, vol. 1, 2nd ed. (Cambridge, Cambridge University Press, 1896), p. 82.

7. Lehmann, “Dreiecke im Dreieck,” pp. 101–20.

CHAPTER 8: TRIANGLE CONSTRUCTIONS

1. Various constructions of the regular pentagon can be found in A. S. Posamentier and I. Lehmann, The Glorious Golden Ratio (Amherst, NY: Prometheus Books, 2012).

2. Gauss used the following relationship:

3. T. Kempermann, Zahlentheoretische Kostproben (Frankfurt am Main, Germany: Harri Deutsch, 2005), p. 35.

4.

5. J. Boehm, W. Börner, E. Hertel, O. Krötenheerdt, W. Mögling, L. Stammler, Geometrie, II. Analytische Darstellung der euklidischen Geometrie (Berlin, Germany: DVW, 1975), pp. 203–205; Walter Gellert, Herbert Kästner, Siegfried Neuber, eds., Lexikon der Mathematik (Leipzig, Germany: Bibliographisches Institut, 1977), pp. 105–106.

6. In order to construct the product and the quotient of two given line segments a and b you can set up the following procedure:

To find the product of the two segments a and b, we use , which gives us x = ab.

To find the quotient of the two segments a and b, we use

CHAPTER 9: INEQUALITIES IN A TRIANGLE

1. ≥ 0 gives us x + y ≥ 2 . With x + y ≥ 2 , x + z ≥ 2 and y + z ≥ 2 we get (y + z)(z + x)(x + y) ≥ 2 . 2 . 2 = 8xyz.

2. This inequality was proposed by Paul Erdös (“Problem 3740,” American Mathematical Monthly 42 [1935]: 396) and solved two years later by Louis Joel Mordell and D.F. Barrow (“Solution to Problem 3740,” American Mathematical Monthly 44 [1937]: 252–54).

3. This is known as the theorem of Möbius-Pompeiu, by the German mathematician and astronomer August Ferdinand Möbius (1790–1868) and the Rumanian mathematician Dimitrie Pompeiu (1873–1954).

4. Leonhard Euler (1707–1783); William Chapple (1718–1781).

5. W. J. Blundon, “Problem E1935,” American Mathematical Monthly 73 (1966): 1122; A. Makowski, “Solution of the Problem E1935,” American Mathematical Monthly 75 (1968): 404.

6. Developed by the Austrian mathematician Roland Weitzenböck (1885–1955).

7. Developed by the Swiss mathematicians Hugo Hadwiger (1908–1981) and Paul Finsler (1894–1970).

CHAPTER 10: TRIANGLES AND FRACTALS

1. A. Farina, S. Giompapa, A. Graziano, A. Liburdi, M. Ravanelli, F. Zirilli, “Tartaglia-Pascal's Triangle: A Historical Perspective with Applications,” Signal, Image and Video Processing (May 24, 2011): 1–16.

2. Kazimir Malevich (1879–1935) was a Russian painter and art theoretician, born of ethnic Polish parents. “He was a pioneer of geometric abstract art and the originator of the Avant-garde Supremacist movement.” (Wikipedia, s.v. “Kazimir Malevich,” last modified May 12, 2012, http://en.wikipedia.org/wiki/Kazimir_Malevich.)

APPENDIX

1. That is, one that can be inscribed in a circle, since in this case, we have the opposite angles supplementary.

2. Ptolemy's theorem states that for a cyclic quadrilateral the product of the diagonals equals the sum of the products of the opposite sides (see p. 274).