who_loves_maths said:
this simple result appears an an exercise problem in the Year 11 3unit (red) Cambridge text.
I thought it was in the year 12 one. Specifically,
Pender, B., et. al., Cambridge Mathematics (3 unit) Year 12, CUP, 2000, Exercise 8C Question 29(c), p 309.
I quite like the 3 unit Cambridge books, especially their "extension" questions.
But it's much older than that. In 1840 a Berlin professor named C. L. Lehmus asked his contemporary Swiss geometer Jacob Steiner for a proof of it. Steiner himself found a proof but published it in 1844. Lehmus proved it independently in 1850. The year 1842 found the first proof in print by a French mathematician:
Lewin, M., On the Steiner-Lehmus theorem, Math. Mag., 47 (1974) 87–89.
There are many other references for it, eg.,:
Sauvé, L., The Steiner-Lehmus theorem, Crux Math., 2 (1976) 19–24.
Trigg, C.W., A bibliography of the Steiner-Lehmus theorem, Crux Math., 2 (1976) 191–193.
I reckon you should be able to find about 80 different proofs!
Here is the list of references given in these 2 articles:
1. Journal fur die reine and angewandte Mathematik, Vol. 28, 1844, p. 376.
2. Archiv der Mathematik and Physik, Vol. 15, 1850, p. 225.
3. J. J. Sylvester, On a simple geometrical problem illustrating a conjectured principle in the theory of geometrical method, Philosophical Magazine, Vol. u, 1852, pp. 366 - 369.
4. Journal de arithematiques elementaires et speciales, 1880, p. 538.
5. F. G.- M., Exercises de Geometrie Mame et Fils, Tours, 1907, pp. 234 - 235.
6. Nathan Altshiller-Court, Mathematical Gazette, Vol. 18, 1934, p. 120.
7. A. Henderson, A classic problem in Euclidean geometry, J. Elisha Mitchell Soc., 1937, pp. 246 - 281.
8. G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1940, P. :111.
9. J. A. McBride, The equal internal bisectors theorem, Edinburgh Mathematical Notes, Vol. 33, 1943, pp. 1 -13.
10. V. Thebault, The Theorem of Lehmus, Scripta Mathematica, Vol. 15, 1949, pp. 87 - 88.
11. Nathan Altshiller-Court, College Geometry, Barnes and Noble, 1952, p. 73.
12. Scientific American, Vol. 204, 1961, pp. 166-168.
13. G. Gilbert and D. McDonnell, The Steiner-Lehmus Theorem, American Mathematical. Monthly, Vol. 70, 1963, pp. 79-80.
14. H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, Random House of Canada, 1967, pp. 14 - 16, 156.
15. H. S. M. Coxeter, Introduction to Geometry, 2nd Edition, Wiley, 1969, pp.9, 420.
16. David C. Kay, College Geometry, Holt, Rinehart and Winston, 1969, pp. 119, 348.
17. J. V. Malesevic, A direct proof of the Steiner-Lehmus Theorem, Mathematics Magazine, Vol. 43, 1970, pp. 101 - 102.
18. Howard Eves, A Purvey of Geometry, Revised Edition, Allyn and Bacon, 1972, pp. 58, 390.
19, K. R. S. Sastry, Problem 862, Mathematics Magazine, Vol. 46, 1973, p. 103.
20. Mordechai Lewin, On the Steiner-Lehmus Theorem, Mathematics Magazine. Vol. 47, 1974, pp. 87 - 89.
21. Lawrence A. Ringenberg, Solution II to Problem 862, Mathematics Magazine, Vol, 47, 1974, p. 53.
22. Charles W. Trigg, Solution I to Problem 862, ibid., pp. 52- 53.
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1. The American Mathematical Monthly, 2 (1895), 157, 189-91; 3 (March 1896), 90; 5 (April 1898), 1o8; 9 (Feb. 1902), 43; 15 (Feb. 1908), 37; 24 (Jan. 1917), 33; 24 (Sept. 1917), 344; 25 (1918), 182-3; 40 (Aug. 1933), 1123.
2. The Mathematics Teacher, 45 (Feb. 1952), 121-2.
3. Mathematical Gazette, Dec. 1959.
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5. Richard Philip Baker, The Problem of the Angle-Bisectors, University of Chicago Press, 98 pages (Circa 1911). O.P.
6. W.E. Bleick, "Angle Bisectors of an Isosceles Triangle," American Mathematical. Monthly, 55 (Oct. 1948), 495.
7. W.E. Buker, (Equal external bisectors), Solution of problem E305, American Mathematical Monthly, 45 (August 1938), 480.
8. Lu Chin-Shih, (Equal external bisectors), Solution of problem 1148, School Science and Mathematics, 31 (April 1931), 465-466.
9. Sister Mary Constantia, "Dr. Hopkins' proof of the angle bisector problem," The Mathematics Teacher, 57 (Dec. 1964), 539-541.
10. J.J. Corliss, "If Two External Bisectors are Equal is the Triangle Isosceles?" School Science and Mathematics, 39 (Nov. 1939), 732-735.
11. N.A. Court, College Geometry, Johnson Publishing Co., (1925), p. 66.
12. A.W. Gillies, A.R. Pargetter, and H.G. Woyda, "Three notes inspired by the SteinerLehmus Theorem," Mathematical Gazette, 57 (Dec. 1973), 336-339.
13. William E. Heal, "Relating to the Demonstration of a Geometrical Theorem," American Mathematical Monthly, 25 (1918), 182-183.
14. Archibald Henderson, "The Lehmus-Steiner-Terquem Problem in Global Survey," Scripta Mathematica, 21 (1955), 223, 309.
15. Archibald Henderson, "A Postscript to an Earlier Article," Scripta Mathematica, 22 (March 1956), 81, 84.
16. Joseph Holzinger, "The Problem of the Angle Bisectors," The Mathematics Teacher, 56 (May 1963), 321-322.
17. L.M. Kelly, (Equal symmedians), Solution of problem E613, American Mathematical Monthly, 51 (Dec. 1944), 590-591.
18. Lady's and Gentleman's Diary (1859), p. 88.
19. Louis Leitner, Harold Grossman, and Joseph Lev. (Equal internal bisectors), Three solutions of problem 1283, School Science and Mathematics, 33 (October 1933), 781-783.
20. C.I. Lubin, "The Theorem of Lehmus and Complex Numbers," Scripta Mathematica, 24 (June 1959), 137-140.
21. David L. MacKay, "The Lehmus-Steiner Theorem," School Science and Mathematics, 39 (June 1939), 561-562.
22. David L. MacKay, "The Pseudo-Isosceles Triangle," School Srience and Mathematics, 40 (May 1940), 464-468.
23. Sharon Murnick, (Equal external bisectors), Solution of problem 34, The Pentagon, 13 (Fall 1953), 35-36.
24. Neilson, Wm. A., Roads to Knowledge, Halcyon House, New York (1941), p. 250.
25. H. Clark Overley, "The Internal Bisector Theorem," School Science and Mathematics, 64 (June 1964), 463-465.
26. Mary Payne, N.D. Lane, and Howard Eves, "The Generalized Steiner-Lehmus Problem," Three treatments of problem E863, American Mathematical Monthly, 57 (January 1950), 37-38.
27. F.A.C. Sevier, "A New Proof of an Old Theorem," The Mathematics Teacher, 45 (Feb. 1952), 121-122.
28. Victor Thebault, "On the isosceles triangle," American Mathematical Monthly, 45 (May 1938), 307-309.
29. Victor Thebault, Solution of problem E339, American Mathematical Monthly, 46 (May 1939), 298-299.
30. Victor Thebault, "Au sujet d'un cas d'egalite des triangles," L'Education Mathematique, 56e Anne, Mars 1954, No. 11, p. 81.
31. Victor Thebault, "Congruent triangles (fifth case) and the theorem of Lehmus," The Mathematics Teacher, 48 (Feb. 1955), 97-98.
32. Victor Thebault, "The Theorem of Lehmus," Scripta Mathematica, 22 (March 1956), 20.
33. Todhunter, The Elements of Euclid, Macmillan, London (1883), 316-317.
34. C.W. Trigg, (Equal n-sectors), Solution of problem 146, National Mathematics Magazine, 12 (April 1938), 353-354.
35. C.W. Trigg, (Equal external bisectors), Solution of problem 224, National Mathematics Magazine, 14 (Oct. 1939), 51-52.
36. C.W. Trigg and G.A. Yanosik, Two solutions of problem E350, American Mathematical Monthly, 46 (Oct. 1939), 513-514.
