Abstract
The theorems of Ceva and Menelaus are concerned with cyclic products of ratios of lengths of collinear segments of triangles or more general polygons. These segments have one endpoint at a vertex of the polygon and one at the intersection point of a side with a suitable line. To these classical results we have recently added a ‘selftransversality theorem’ in which the ‘suitable line’ is determined by two other vertices. Here we present additional ‘transversality’ properties in which the ‘suitable line’ is determined either by a vertex and the intersection point of two diagonals, or by the intersection points of two pairs of such diagonals. Unexpectedly it turns out that besides several infinite families of systematic cases there are also a few ‘sporadic’ cases.
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References
Baptist, P.: Die Entwicklung der neueren Dreiecksgeometrie, BI Wissenschaftsverlag, Mannheim, 1992.
Carnot, L. N.M.: Géometrie de position, Duprat, Paris, 1803.
Chou, S.-C., Gao, X.S. and Zhang, J.Z.: Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry Theorems, World Scientific, Singapore, 1994.
Crelle, A. L.: Ñber einige Eigenschaften des ebenen geradlinigen Dreiecks rücksichtlich dreier durch die Winkel-Spitzen gezogenen geraden Linien, Berlin, 1816.
Grünbaum, B. and Shephard, G. C.: Ceva, Menelaus, and the area principle, Math. Magazine 68 (1995), 254–268.
Grünbaum, B. and Shephard, G. C.: Ceva, Menelaus and selftransversality, Geom. Dedicata 65 (1997), 179–192.
Grünbaum, B. and Shephard, G. C.: A new Ceva-type theorem. Math. Gazette 80 (1996), 492–500.
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Grünbaum, B., Shephard, G.C. Some New Transversality Properties. Geometriae Dedicata 71, 179–208 (1998). https://doi.org/10.1023/A:1005069213744
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DOI: https://doi.org/10.1023/A:1005069213744