## Study of the Cevians and Extension of the Stewart’s Theorem by Nayoung Ko

May 13, 2020

Abstract

Cevians are the lines inside a triangle. These three cevians form six triangles and with this information, this research is to extend the knowledge of cevians through Stewart’s Theorem. By using Stewart’s Theorem, the relationship between transversal and the sides of a trapezoid was also found. After deriving Stewart’s Theorem, the relationship under special case was able to be analyzed. It was possible to fit the existing formula into more various and interesting cases such as two cevians and two intersecting cevians. This project was especially significant because we discovered that it was possible to derive formulas with n-cevians from the same vertex by using the formerly derived equation for (n-1) cevians. In other words, we can derive an equation with three cevians from the same vertex by using the equation for two cevians from the same vertex. For further study, we can extend Stewart’s Theorem to apply to other polygons which will then relate the perimeter of the polygon to areas of triangles in the polygon.