Modification of the Inner Cross Theorem on Crossed Quadrilateral

Nour Fitri; Mashadi; Sri Gemawati

doi:https://doi.org/10.14445/22315373/IJMTT-V69I2P507

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 69 | Issue 2 | Year 2023 | Article Id. IJMTT-V69I2P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I2P507

Modification of the Inner Cross Theorem on Crossed Quadrilateral

Nour Fitri, Mashadi, Sri Gemawati

Received	Revised	Accepted	Published
26 Dec 2022	31 Jan 2023	12 Feb 2023	20 Feb 2023

Citation :

Nour Fitri, Mashadi, Sri Gemawati, "Modification of the Inner Cross Theorem on Crossed Quadrilateral," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 2, pp. 53-60, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I2P507

Abstract

This article discusses the inner development of Cross's Theorem on crossed quadrilaterals. By doing two steps, namely expanding the square leads inward once and twice. The proof is based on the rule of sines and the rule of cosines. The achieved result for a one-time expansion is \ |LΔBFG - LΔDKJ| = L◻ABCD and |LΔLAE - LΔICH| = L ◻ABCD and for double expansion, the difference between L ◻EFON and L ◻IJSR is equal to 2L◻ABCD and the difference between L KLMT and L GHQP is zero.

Keywords

Crossed Quadrilateral, Sine and Cosine, Cross Theorem.

References

[1] G. Faux, “Happy 21st Birthday Cockcroft 243 and All the Other Threes,” Mathematics Teaching, vol. 189, pp. 10-12, 2004.
[2] L. Baker, and I. Harris, “A Day to Remember Kath Cross,” Mathematics Teaching, vol. 189, pp. 20-22, 2004.
[3] J. Gilbey, “Responding to Geoff Faux’s Challenge,” Mathematics Teaching, vol. 190, pp. 16, 2005.
[4] Mashadi, “Advanced Geometry II, Pekanbaru,” UR Press, pp. 301-307, 2020.
[5] Wolfran Deminstrations Project, 2017. [Online]. Available: http://demonstrations.wolfram.com/Crosss
[6] Michael De Villiers, “An Example of the Discovery Function of Proof,” Mathematics in School, vol. 36, no. 4, pp. 9-11, 2007. Crossref, http://dx.doi.org/10.2307/30216041
[7] Manuel Luis, Students Development of Mathematical Practices Based on the Use of Computational Technologies, Center for Research and Andvanced Studies, Mexicos 2006.
[8] M. Rusdi Syawaludin, Mashadi Mashadi, and Sri Gemawati, “Modification Cross’ Theorem on Triangle with Congruence,” International Journal of Theoretical and Applied Mathematics, vol. 4, no. 5, pp. 40-44, 2018. Crossref, http://dx.doi.org/10.11648/j.ijtam.20180405.11
[9] Saniyah, Mashadi, and Sri Gemawati, “Modification of the Cross Theorem on Non-Convex Quadrilateral,” International Journal of Mathematics Trends and Technology, vol. 68, no. 7, pp. 43-51, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I7P507
[10] Michael De Villiers, “Slaying a Geometrical Monster: Finding The Area of a Crossed Quadrilateral,” Learning and Teaching Mathematics, no. 18, pp. 23-28, 2015.
[11] Michael De Villiers, “A Sketchpad Discovery Involving Triangles and Quadriliteral,” KZN Mathematics Journal, vol. 28, pp. 18-21, 2012.
[12] Mashadi, Teaching Mathematics. Pekanbaru, UR Press, pp. 86-97, 2015.
[13] Mashadi, Geometry Advanced, Pekanbaru, UR Press, pp. 126-131, 2015.
[14] Amelia, Mashadi, and Sri Gemawati, “Alternatif Proofs for the Lenght of Angle Bisector Theorem on Triangle,” International Journal of Mathematics Trends and Technology, vol. 66, no. 10, pp. 163-166, 2020. Crossref, https://www.ijmttjournal.org/archive/ijmtt-v66i10p519
[15] S. Lang, and G. Murrow, Geometry Second Edition, Springer-Verlag, New York, 1988.
[16] Mashadi, Chitra Valentika, and Sri Gemawati, “Developtment of Napoleon on the Rectangles in Case of Inside Direction,” International Journal of Theoretical and Applied Mathematics, vol. 3, no. 4, pp. 54-57, 2017.
[17] Chitra Valentika et al., “The Development of Napoleons Theorem on the Quadrilateral in Case of Outside Direction,” Pure and Applied Mathematics Journal, vol. 6, no. 4, pp. 108-113, 2017.
[18] Chitra Valentika, Mashadi, and Sri Gemawati, “The Development of Napoleons Theorem on Quadrilateral with Congruence and Trigonometry,” Bulletin of Mathematics, vol. 8, no. 01, pp. 97-108, 2016.
[19] M. Manalu, M. Mashadi, and S. Gemawati, “Development of Van Aubel's Theorem on Cross Quadrilaterals”, PRISMA, Proceedings of the National Mathematics Seminar, Vol. 5, pp. 850-860, 2022.
[20] Ariska, Mashadi, and Leli Deswita, “Modification of Varignon's Theorem,” International Journal of Mathematics and Computer Research, vol. 9, no. 12, pp. 2526–2529, 2021. Crossref, https://doi.org/10.47191/ijmcr/v9i12.03
[21] Y. Hartati, and Mashadi, “Circumcenter Point Triangle Modification of Napoleon's Theorem,” Pattimura Proceeding: Conference of Science and Technology, vol. 2, no. 1, pp. 67-76, 2022.
[22] M. Mulyadi et al., “Development of Van Aubel's Theorem on Hexagons,” Journal of Mathematical Paedagogy, vol. 1, no. 2, pp. 119- 128, 2017.
[23] C.E. Garza-Hume, Maricarmen C.Jorge, and Arturo Olvera, “Areas and Shapes of Planar Irregular Polygons,” Forum Geometriorum, vol. 18, pp. 17-36, 2018.
[24] John Michael Rassias, “Euler Type Theorems on Quadrilaterals Pentagons and Hexagons,” Mathematical Sciences Research Journal, vol. 10, no. 8, pp. 196, 2006.
[25] H. S. M. Coxeter, and Samuel L. Greitzer, Geometry Revisited, Washington D.C., The Mathematical Association of America, 1967.