How Does One Cut a Triangle?
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Including dozens of proofs and counterexamples, this second edition of Soifer’s inspirational book uses geometry, algebra, trigonometry, linear algebra, and rings to show how different areas of mathematics can be juxtaposed in the solution of a given problem.
mathematics, apt to bring pleasure to anybody willing to devote a few hours to follow its adventures among solved and unsolved questions. Branko Grunbaum ¨ Professor of Mathematics University of Washington January 1990, Seattle, Washington xv There is a view, held by many, that mathematics books are dull. This view is not without support. It is reinforced by numerous examples at all levels, from elementary texts with page after page of mindnumbing drill to advanced books written in a
map parallel lines into parallel lines. Problem 8.4.8. Parallel projections preserve the ratios of lengths of collinear segments. 82 8 Convex Figures and the Function S( F ) L p P′ f ( p) P Figure 8.14 Problem 8.4.9. Parallel projections preserve the ratios of areas. Problem 8.4.10. (I. Yaglom [Y4], Vol. 3, p. 13 of the English translation) Let A, B, C be three non-collinear points in a plane P, and let M, N, K be three non-collinear points in a plane P′ . Then the planes P and P′ can be
Boltyanski–Soifer book [BS], and finds the first lower bound: 13 Matthew Kahle on the Five-Point Problem 141 Lower Bound 13.3. [Kah] α ≥ 1/6. Proof. Behold (Figure 13.1): Figure 13.1 Points b and e are midpoints of the sides, while c and d partition the base into three equal parts. Triangles abc, ade, bcd, and cde all have area exactly 1/6. ⊔ ⊓ Of course, Kahle’s Theorem provides the upper bound: α ≤ 6/25. 142 13 Matthew Kahle on the Five-Point Problem Now it is time for conjecturing
Math. Magazine 52(1) (1979), 131–145. Laczkovich, M., Equidecomposability and Discrepancy: A Solution of Tarski’s Circle-Squaring Problem, J. fur ¨ die Reine und Angewandte Math. 404 (1990), 77–117. References 169 Laczkovich, M., Tilings of triangles, Discrete Math. 140 (1–3) (1995), 79–94. [L3] Laczkovich, M., Tilings of Polygons with Similar Triangles, II, Discrete Comput. Geom. 19 (1998), 411–425. [L4] Laczkovich, M., On the Number of Pieces in Tilings of Triangles, A note written for this
thrilled about it. And yet, in a way, I was satisfied. The great Kolmogorov thought Grand II was a difficult problem. “What would he think then about Grand I?” I exclaimed to myself. Nineteen years later, Grand II has finally made it into the Mathematical Olympiad of the International Summer Institute at Oakdale, Long Island, New York. The competitors included some remarkable high school students from the Soviet Union, France, Switzerland, and the United States. The winner, Vania Arzhantsev, a