## Study on the Geometric Properties in the Cevasix Triangle

#### Abstract:

#### Bibliography/Citations:

[1] S. Abbot, Average sequences and triangles, Mathematical Gazette, 80 (1996), 222–224.

[2] G. Z. Chang and P. J. Davis, Iterative processes in elementary geometry, American Mathematical Monthly 90 (1983), no. 7, 421-431.

[3] G. Z. Chang and T. W. Sederberg, Over and over again, New Mathematical Library, no. 39, Mathematical Association of America, Washington DC, 1997.

[4] R. J. Clarke, Sequences of polygons, Mathematics Magazine 90 (1979), no 2, 102-105.

[5] H. S. M. Coxeter and S. L. Greitzer, Geometry revisited, Mathematical Association of America, Washington D.C., 1967, 22-26.

[6] H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved problems in geometry, Problem books in mathematics - Unsolved problems in intuitive mathematics, II, Springer-Verlag, New York, 1994.

[7] P. J. Davis, Cyclic transformations of polygons and the generalized inverse, Canadian Journal of Mathematics 29 (1977), no. 4, 756-770.

[8] P. J. Davis, Circulant matrices, Wiley-Interscience Publication, Pure and Applied Mathematics, John Wiley and Son, New York, Chichester, Brisbane, 1979.

[9] L. R. Hitt and X. M. Zhang, Dynamic geometry of polygons, Elemente der Mathematik, 56 (2001), no.1, 21-37.

[10] D. Ismailescu and J. Jacobs, On sequences of nested triangles, Periodica Mathematica Hungarica, 52 (2006), 169-184

[11] P. T. Krasopoulos, Kronecker’s Approximation theorem and a sequence of triangles, Forum Geometricorum, 8 (2008), 27–37.

[12] B. Ziv, Napoleon-like configurations and sequences of triangles, Forum Geometricorum, 2 (2002), 115-128.

[13 ]“Stewart, Matthew.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. 28 Dec. 2013 http://www.encyclopedia.com.

[14]“Apollonius of Perga.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. 28 Dec. 2013 http://www.encyclopedia.com.

[15]“Stewart's Theorem.” AoPSWiki. Art of Problem Solving, 28 Dec. 2011. Web. 29 Dec. 2013.

#### Additional Project Information

#### Research Plan:

In mathematics, a triangle is the most basic and important 2-dimensional closed domain which can be used for academic use, industrial design, and structural engineering. It is interesting to extend existing theorems to apply to different triangles and other polygons which will then relate to each other.

After deriving Stewart's Theorem, the relationship under special cases was able to be analyzed, and thus it was possible to extend to more various and interesting cases with more variables than two cevians. It will be possible to derive formulas with n-cevian from the same vertex by using the formerly derived equation for (n-1) cevians. In other words, it is possible to derive an equation with three cevians from the same vertex by using the equation for two cevians and two intersecting cevians from the same vertex. Also, it is interesting to extend Stewart’s Theorem to other polygons to relate the perimeter of the polygon to areas of triangles in the polygon.

In this research, the relationship between the cevians, sides of a triangle, and the segments it forms on the opposite side were studied. After deriving Stewart’s Theorem in a different way, I delved more with amplifying knowledge to learn that the existing equation of Stewart’s Theorem could be modified if a triangle had more than one cevian. Special circumstances such as Apollonius’ Theorem, angle bisector, altitude, isosceles triangle, and equilateral triangle were useful because they could make the equation simplest form.

Although it is challenging to simplify the calculations of two cevians and two intersecting cevian, these unique cases can further support the versatile characteristics of the theorem. For further consideration, I aspired to study the relationship between the perimeter of the hexagon formed by connecting six centroids and the perimeter of a starting triangle. With a firm understanding of a cevasix triangle, it will be interesting and also challenging to explore geometric shapes such as hexagons inside a triangle. Thereafter a new interesting theorem, a trigonometric form of Ceva’s Theorem, was finally found.

#### Questions and Answers

1. What was the major objective of your project and what was your plan to achieve it?

a. Was that goal the result of any specific situation, experience, or problem you encountered?

b. Were you trying to solve a problem, answer a question, or test a hypothesis?

The objective of this project was to develop a new mathematical theory that could be applied to engineering or design. Geometry has always been my field of interest as all points, lines, and two-dimensional figures are inevitably related to each other in an interesting manner. In this project, I was mostly trying to analyze and solve a problem. After deriving Stewart's Theorem, I extended the formula to various cases to find its relationship with different numbers of cevians and polygons.

2. What were the major tasks you had to perform in order to complete your project?

a. For teams, describe what each member worked on.

First, I had to review the formulas related to cevians and cevasix configurations to find the relevant correlations. Then, to extend on it, I had to modify them using the Apollonius' Theorem and basic properties of isosceles and equilateral triangles. The last step was to derive formulas for special situations and extend the work I did.

3. What is new or novel about your project?

a. Is there some aspect of your project's objective, or how you achieved it that you haven't done before?

b. Is your project's objective, or the way you implemented it, different from anything you have seen?

c. If you believe your work to be unique in some way, what research have you done to confirm that it is?

My project is novel because it delves deep into a concept that seems basic and extends on it. I have made new attempts such as deriving formulas with n-cevians from the same vertex and finding the relationship between the perimeter of the polygon and the areas of triangles in the polygon.

4. What was the most challenging part of completing your project?

a. What problems did you encounter, and how did you overcome them?

b. What did you learn from overcoming these problems?

The most difficult part of the project was the process of getting to the final answer with multiple variables. For example, the most difficult theorem to prove was two cevians from the same vertex because this was an extended form of one cevian and all calculations had to be accurate in order to apply this same equation to three cevians from the same vertex.

5. If you were going to do this project again, are there any things you would do differently the next time?

In mathematics, a triangle is the most basic and important 2 dimensional closed domain which can be used for academical use, industrial design and many structural engineering. I was interested in studying further to extend existing theorems to apply the basics to different triangles and other polygons which will then relate to each other.

6. Did working on this project give you any ideas for other projects?

Yes, I thought that it'll

7. How did COVID-19 affect the completion of your project?

There were no specific challenges caused by COVID-19.