Volume 68 | Issue 6 | Year 2022 | Article Id. IJMTT-V68I6P526 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I6P526

Received | Revised | Accepted | Published |
---|---|---|---|

27 May 2022 | 04 Jul 2022 | 07 Jul 2022 | 14 Jul 2022 |

This article discusses the modification of the Japanese on Heptagon. The proof is done using Carnot’s Theorem, the modification of Ptolemy’s Theorem, and the Japanese Theorem. The result obtained is the proof using Carnot;s Theorem is more effective. So, the sum of the radius length of the incircle of the triangles with diagonal lines formed from all points is the same.

[1] W. J. Greenstreet, “Japanese Mathematics,” The Mathematical Gaxette., vol. 3, no. 55, pp. 268-270, 1906.

[2] R. Honsberger, Mathematical Gem III. New York, Mathematical Association of America, pp. 24-26, 1985.

[3] W. Reyes, “An Application of Thebault’s Theorem,” Forum Geometricorum, vol. 2, no. 12, pp. 183-185, 2002.

[4] M. Ahuja, W. Uegaki, and K. Matsushita, “Japanese Theorem: A little known Theorem with Many Proofs – Part II”, Missouri Journal of Mathematical Science, vol. 16, no. 3, pp. 149-158, 2004.

[5] N. Minculete, C. Barbu, and G. Szollosy, “About the Japanese Theorem,” Crux Mathematicorum, vol. 38, no. 5, pp. 188-193, 2012.

[6] M. Ahuja, W. Uegaki, and K. Matsushita, “Japanese Theorem: A little known theorem with many proofs – part I”, Missouri Journal of Mathematical Science, vol. 16, no. 2, pp. 72-81, 2004.

[7] Mashadi, Geometri, Pekanbaru, UR Press, pp. 166-226, 2015.

[8] Mashadi, Geometri Lanjut. Pekanbaru, UR Press, pp. 37-96, 2015.

[9] F. Perrier, “Carnot’s Theorem in Trigonometric Disguise,” The Mathematical Association, vol. 91, no. 520, pp. 115-117, 2007.

[10] A. Claudi and B. N. Roger, “Proof without words: Carnot’s Theorem for acute triangles,” The College Mathematics Journal, vol. 39, no. 2, pp. 111, 2010.

[11] S. Lang and G. Murrow, Geometry 2nd ed., New York, Spring-Verlag, pp. 163, 2000.

[12] H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, Washington D.C., “The Mathematical Association of America,” pp. 56-57, 1967.

[13] N. A. Court, “An Introduction to the Modern Geometry of the Triangle and Circle,” New York, Dover Publication, Inc., pp. 127-129, 2007.

[14] R. A. Johnson, “Advanced Euclidean Geometry,” New York, Dover Publication, Inc., pp. 85, 1985.

[15] G. W. I. S. Amarasinnghe, “A Concise Elementary Proof for the Ptolemy’s Theorem,” Global Journal of Advanced Research on Classical and Modern Geometris, vol. 2, no. 1, pp. 20-25, 2010.

[16] D. N. V. Krishna, “The New Proof of Ptolemy’s Theorem and Nine-Point Circle Theorem,” Mathematics and Computer Science, vol. 1, no. 4, pp. 93-100, 2016.

Nonong Wahyuni, Mashadi, Sri Gemawati, "Modification of the Japanese Theorem on Heptagon," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 68, no. 6, pp. 205-210, 2022. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V68I6P526