This paper proves Euclid’s fifth postulate and convergence of straight lines using the formula for the area of trapezoids and assuming straight lines, it derives a general formula for the area of trapezoids involving ratios and we assume that the straight lines determine the nature and area for all the rectilinear figures. Furthermore, this proof is essential in Geometric optics basically in proving and classifying beams of light (wave) that is to mathematically prove the presence of parallel, convergent and divergent beams of light assuming the ray of light is a straight line.
References
[1]
Greenberg, M.J. (2008) Euclidean and Non-Euclidean Geometries-Development and History. Frontiers in Mathematics. W. H. Freeman, New York.
[2]
Diaz, J.E. (2018) Fifth Postulate of Euclid and the Non-Euclidean Geometries. International Journal of Scientific and Engineering Research, 9, 530.
[3]
Rosenfeld, B.A. (2012) A History of Non-Euclidean Geometry. Springer-Verlag New York Inc., New York.
[4]
Euclid, J.L. and Fitspatrick, R. (2008) Euclid’s Elements of Geometry. Fitspatrick, R., Ed. University of Texas, Austin.
[5]
Allen, F.B. (1960) School Mathematics Study Group—Geometry Part II. Yale University Press, New Haven.
[6]
Salilew, G.A. (2017) New Approach for Similarity of Trapezoids, Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 17, 2249-4626.
